Alain Goriely and Michael Tabor- The nonlinear dynamics of filaments

8/3/2019 Alain Goriely and Michael Tabor- The nonlinear dynamics of filaments

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The nonlinear dynamics of filaments

Alain Goriely and Michael Tabor

University of Arizona

Program in Applied Mathematics,

Building #89, Tucson, AZ85721, USA

e-mail: goriely at math.arizona.edu

August 10, 1998

Abstract

The Kirchhoff equations provide a well established framework to study the statics and

dynamics of thin elastic filaments. The study of static solutions to these equations has along history and provides the basis for many investigations, both past and present, of theconfigurations taken by filaments subject to various external forces and boundary conditions.Here we review recently developed techniques involving linear and nonlinear analyses thatenable one to study, in some detail, the actual dynamics of filament instabilities and thelocalized structures that can ensue. By introducing a novel arc-length preserving perturbationscheme a linear stability analysis can be performed which, in turn, leads to dispersion relationsthat provide the selection mechanism for the shape of an unstable filament. These dispersionrelations provide the starting point for nonlinear analysis and the derivation of new amplitudeequations which describe the filament dynamics above the instability threshold. Here we willmainly be concerned with the analysis of rods of circular cross-sections and survey the behaviorof rings, rods, helices and show how these results lead to a complete dynamical description offilament buckling.

1 Introduction

The classic buckling instability of a thin elastic filament when it is subjected to the appropriateamount of twist, lies at the heart of many important processes ranging from the kinking of tele-phone cables when they are being laid (Zajac, 1962; Coyne, 1990), to the super-coiling propertiesof long molecular structures such as proteins, DNA and bacterial fibers (Barkley & Zimm, 1979;Benham, 1985; Mendelson, 1990; Thwaites & Mendelson, 1991; Hunt & Hearst, 1991; Schlick &Olson, 1992; Yang et al., 1993; Bauer et al., 1993; Shi & Hearst, 1994). In addition, long, thin,twisted elastic structures play important paradigmatic roles in the theory of polymers and liquidcrystals (Goldstein & Langer, 1995; Shelley & Ueda, 1996); characterizing the motion of vortextubes in hydrodynamics (Keener, 1990); and modeling the formation of sun spots and heating ofthe solar corona (Spruit, 1981; Silva & Chouduri, 1993). These applications have spawned manytheoretical and computational studies and have also prompted the development of experimentalinvestigations to help bench-mark the theoretical work. These experimental studies range fromthe controlled buckling studies of macroscopic elastic rods (Thompson & Champneys, 1996) tothe ingenious manipulations of DNA strands (Schlick, 1995).

A filament is defined, roughly speaking, to be a three dimensional body whose cross-sectionis much smaller than its length. If, furthermore, the curvature of the filament is assumed to besmall relative to its length, it is possible to show that its dynamics is governed by the celebratedKirchhoff equations. (An excellent historical study of these equations is provided by Dill (Dill,

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1992)). These equations describe the evolution of a filament in (three-dimensional) space andconsist of a system of six coupled, nonlinear PDEs of second-order in time and arc-length; thelatter variable providing the intrinsic coordinatization of the filament.

A considerable amount of work has been devoted to investigating the stability of given sta-tionary filament configurations (Love, 1892; Landau & Lifshitz, 1959; Timoshenko & Gere, 1961)through the study of the static Kirchhoff equations - which form a much more tractable system of

six coupled nonlinear ODEs. As is well known, these are exact analogues of the Euler equationsfor rigid-body motion and, accordingly, are often cast in terms of the Euler angles to characterizethe orientation of the local basis describing the twisting and bending of the rod (as a function ofarc-length). This type of analysis has yielded a very a rich variety of exact solutions from whichconclusions about the stability are often drawn.

Analytical study of the full dynamical equations is a very challenging problem and to datehas mainly been restricted to the study of traveling wave solutions (Coleman & Dill, 1992). Analternative approach has been to introduce a time-dependent Lagrangian to describe elementaryvibrations of the equilibrium solutions (Tanaka & Takahashi, 1985; H., 1987). However, theseLagrangians are ad-hoc models which are not related to the dynamical Kirchhoff equations anddo not truly describe the dynamics of real elastic filaments. An alternative to analytical work is

provided by numerical simulation - a nontrivial task in its own right - and this approach has beenpursued with some success by a number of groups (Maddocks & Dichmann, 1994), (Klapper &Tabor, 1996; Kleckner, 1996).

In order to obtain a complete, dynamical, picture of filamentary instabilities we depart fromtraditional approaches and consider the linear stability of static solutions to the full, three-dimensional, dynamical Kirchhoff model: this is achieved through the development a new arc-length preserving perturbation scheme. The dispersion relations that ensue from the associatedlinear problem, then provide the starting point for a globalweakly nonlinear analysis which pro-vides a description of the evolution of unstable structures beyond threshold.

The structure of this review paper is as follows. In Section 2, we briefly review the derivationof the Kirchhoff equations in order to introduce the basic notation. In Section 3, we introduce theperturbation scheme which is developed in terms of the so-called director basis. The fundamental

property of this perturbation expansion is that it preserves arc-length (and hence total filamentlength) at each order of the perturbation. Applying this scheme to the Kirchhoff equations yieldsvariational equations governing the stability of stationary solutions. This approach is illustratedin Section 4 with the examples of the twisted planar ring, the twisted straight rod, and a helicalrod. In Section 5 we describe our nonlinear analysis of the Kirchhoff equations and illustrate thetechnique for the circular ring and the straight rod. The nonlinear analysis of the ring is performedat the level of a single mode, which leads to a nonlinear amplitude equation in the form of a singleordinary differential equation. In the case of the twisted straight rod, which involves a continuumof modes, the analysis leads to amplitude equations which take the form of a pair of couplednonlinear Klein-Gordon equations. The nonlinear analysis provides vital information about theamplitude of the structures that appear after bifurcation, and is essential for the description of the

localization that occurs after buckling. In the remaining three sections of this review we describea variety of applications of our approach to filament problems. In section 6 we give a summaryof our dynamical model of buckling of a straight rod subjected to twisting forces; and in Section7 we describe the fundamental role played by intrinsic curvature and twist in the morphogenesisof naturally occurring filaments. Although the earlier parts of this review follow, fairly closely,the presentations given in our recent series of papers (Goriely & Tabor, 1996; Goriely & Tabor,1997a; Goriely & Tabor, 1997b; Goriely & Tabor, 1997c; Goriely & Tabor, 1998a), a variety ofnew, or previously unpublished, results are sprinkled throughout, especially in the later sections.

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2 The Kirchhoff model

A fundamental idea in Kirchhoffs theory of thin filaments is to represent all the physically relevantstresses, i.e. the forces and the moments, as cross-sectional averages at each point along the spacecurve representing the axis of the filament. It is this that enables one to write down a spatiallyone-dimensional model for a filament.

Let x = x(s, t) :RRR3

be the space curve, parameterized by the arc length s, representingthe filament axis, whose position can also depend on the time t. The curve is assumed to be atleast three times differentiable with respect to s. For each value of s and t, a local, orthonormalcoordinate system, called the director basis, {d1, d2, d3} is associated with the curve by identifyingthe vector d3(s, t) = x

(s, t) as the tangent vector of x at s (the prime denotes the s-derivative),and taking the vectors d1, d2 to span the plane normal to d3, such that {d1, d2, d3} forms a right-handed triad (d1 d2 = d3, d2 d3 = d1). In the case where d1 is along d3, the director basisspecializes to the well-know Frenet triad where d1 is the normal vector and d2 the bi-normalvector. The director basis is more general (and more useful) than the Frenet basis since thevectors {d1(s, t), d2(s, t)} can be chosen to coincide with the principle axes of the rod cross-sectionand hence capture the actual twist of the rod itself (rather than just the torsion of the axial spacecurve).

Given the director basis {d1, d2, d3}, the filament geometry can be reconstructed at all timesby integrating the tangent vector, i.e. x(s, t) =

sd3(s, t)ds.

The kinematics of thin filament evolution can be easily written in term of the director basis.Taking into account the orthonormality of this basis, one readily obtains its evolution with respectto arc length and time; namely

di =

3j=1

Kijdj i = 1, 2, 3, (1.a)

di =3

j=1Wijdj i = 1, 2, 3, (1.b)

where ( ) stands for the time derivative. W and K are the antisymmetric 3 by 3 matrices:

K =

0 3 23 0 1

2 1 0

, W =

0 3 23 0 1

2 1 0

. (2)

The elements of K make up the components of the twist vector, namely =3

i=1 idi; and theelements of W define the components of the spin vector, namely =

3i=1 idi. The two linear

system (1) must be compatible: thus by cross-differentiation one determines that:

W

K = [W, K] , (3)

where [., .] is the matrix commutator: [W, K] = W.KK.W

To go from the kinematics of the director basis to the actual dynamics of an elastic rod, wenow consider the forces and moments acting in such a rod (here with a circular cross section) withthe axial space curve x = x(s, t). The main physical assumptions considered here are: no sheardeformation, no axial extensibility, linear constitutive relationships and circular cross sections (thecase of non-circular cross-sections is considered in detail in (Goriely et al., 1999). As indicatedabove, the stresses are averaged over the cross sections along (and normal to) the central axis:this enables the total force F = F(s, t) and moment M = M(s, t) to be expressed locally in term

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of the director basis, i.e. F =3

i=1 fidi, M =3

i=1 Midi. Conservation of linear and angularmomentum then provides the fundamental equations of motion (Coleman et al., 1993):

F = Ad3, (4.a)

M + d3 F = I(d1 d1 + d2 d2), (4.b)where I is the moment of inertia (about a radial cross section), is the density and A the area

of a (circular) cross section.The equations are completed by the constitutive relationship oflinearelasticity theory; namely

a relationship between the moment and the strains of the form:

M = EI

(1 (u)1 )d1 + (2 (u)2 )d2

+ 2I(3 (u)3 )d3, (5)

where E is the Young modulus and the shear modulus and the (u)i are the intrinsic curva-

tures of the filament. For the meantime we will set these to zero (which implies that the rod isnaturally straight and untwisted); but we will later discuss how they can provide the fundamen-tal driving force for certain types of spontaneous morphogenesis, e.g supercoiling of bacterialfilaments (Goriely et al., 1998) and the helix-hand reversal seen in the tendrils of climbing plants

(Goriely & Tabor, 1998b).The standard scalings

t t

I/AE s s

I/A,

F AEF M ME

AI,

A/I

AE/I. (6)

reduce Eqs. (4-5), to the dimensionless form:

F = d3, (7.a)

M + d3 F = d1 d1 + d2 d2, (7.b)

M = 1d1 + 2d2 + 3d3, (7.c)where = 2/E = 1/(1 + ) characterizes the elastic property of the filament: varying between

2/3 (incompressible case) and 1 (hyper-elastic case). We also note that in the chosen scalings thecharacteristic length scale is set by the filament cross section.

By inserting the constitutive relationship for M (7.c) in (7.b) one obtains explicit differentialrelationships between the lateral forces and the strains. The tangential component of the force,i.e. f3, corresponds to the tension (or compression) in the filament and this must be obtained fromthe condition that d3 maintains its unit norm. Together with the twist and spin equations (1),one obtains, overall, a system of 9 equations (the Kirchhoff equations) for 9 unknowns (f , ,) =(f1, f2, f3, 1, 2, 3, 1, 2, 3) which we write in the shorthand form:

E(f , ,; s, t) = 0. (8)

If one fixes a direction in space, the local vectors , can be expressed in terms of the Euler anglesand the system (8) can then be written as a set of 6 equations for 6 unknowns. However, it wasshown in (Goriely & Tabor, 1996) that the Euler angles are not always the best choice of variablesfor studying the stability of stationary solutions. This was illustrated by an analysis of a twistedplanar ring: standard statics in the Euler basis yields the well known result that the critical twistdensity needed to induce buckling in a ring of radius r is c =

3/r; but calculation of the

energy (to third order) and the action (to fourth order) did not reveal whether the buckled stateis, in fact, the preferred one relative to the initial state. Calculations such as these motivated usto develop a perturbation scheme in terms of the director basis itself.

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3 Perturbation Scheme

The basic idea is to expand the director basis associated with the deformed configuration of thefilament around the basis of the unperturbed stationary solution, and impose the condition thatit remains orthonormal at each order in the perturbation parameter. Namely:

di = d(0)

i+ d

(1)

i+ 2d

(2)

i+ ... i = 1, 2, 3, (9)

Applying the orthonormality condition di.dj = ij leads to an expression for the perturbed basisin terms of the unperturbed basis:

d(1)i =

3j=1

A(1)ij d

(0)j , (10.a)

d(2)i =

3j=1

A(2)ij + S

(2)ij

d(0)j , (10.b)

...

d(n)i =

3j=1

A(n)ij + S

(n)ij

d(0)j , (10.c)

where A(k) is the antisymmetric matrix:

A(k) =

0

(k)3 (k)2

(k)3 0 (k)1(k)2 (k)1 0

, (11)

and S(k) is a symmetric matrix whose entries depend only on (j)i with j < k. For example:

S(2) =

12((1)2 )2 12((1)3 )2 12(1)1 (1)2 12(1)1 (1)312(1)1 (1)2 12((1)3 )2 12((1)1 )2 12(1)2 (1)312

(1)1

(1)3

12

(1)2

(1)3 12(

(1)1 )

2 12((1)2 )

2

, (12)

Once the vector (1) is known the configuration of the perturbed rod is easily reconstructed byintegrating the tangent vector:

x(s, t) =

sds

d(0)3 + (

(1)2 d

(0)1 (1)1 d(0)2 )

+ O(2). (13)

Since any local vector V =

3i=1 vidi can be expanded in terms of the perturbed basis; namely

V = V(0) + V(1) + 2V(2) + ..., we can write a perturbative expansion of the twist and spinmatrices, i.e. K = K(0) + K(1) + ..., W = W(0) + W(1) + ..., where

K(1) =

sA(1) +

A(1), K(0)

, (14.a)

W(1) =

tA(1) +

A(1), W(0)

. (14.b)

If needed higher order terms in the expansion can be generated and easily expressed in terms ofthe lower order terms.

Using these equations, one can write the first order perturbation of the Kirchhoff equations (7.a,7.b) in terms of ((1), f(1)). This system will be referred to as the dynamical variational equations.

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Its solutions control the stability, or lack thereof, of the stationary solutions with respect to lineartime-dependent modes. To emphasize the linear character of these equations we rewrite them asa linear system of 6 equations for the 6-dimensional vector (1) = ((1), f(1)):

LE((0), f(0)).(1) = 0, (15)

where LE is a second-order differential operator in s and t whose coefficients depends on s throughthe unperturbed solution ((0), f(0)).

4 Linear Stability Analysis

The linear system (15) can be used to determine stability in the standard way: for a given staticconfiguration (characterized by the ((0), f(0))), stability to linear disturbances is determined bysetting the components of to be of the form:

j = et

Axjeins/L + Axje

ins/L

j = 1, ...., 6 (16)

where ( ) stands for the complex conjugate, n is the mode number, and L the filament length.(Forinfinite rods, n/L is replaced by the continuous wave number n). At the level of the linear theorythe amplitude, A, is arbitrary; it can only be determined by proceeding to the nonlinear analysisdescribed in Section V.

The values of the growth rates for which (16) satisfies the variational equations (15), arethe ones for which = det(L) = 0. = 0 defines the dispersion relation. This fundamentalexpression relates the mode number n to the growth rate . The technically difficult part of thiscalculation is computational: the dispersion relations are high order polynomials in and n andcan involve many hundreds of terms (for the helix, it is over 500 terms!) and can really only behandled by substantial symbolic manipulation. We note that in all the linear stability analyseswe have studied to date, we only consider R. The dispersion relation can have other solutionscorresponding to vibration modes (i.e. i

R)). These modes are not spontaneously unstable,

i.e. their amplitude is at most of the size of the perturbation itself and they would have to beexplicitly excited in order to be observed. Although such oscillatory modes are apparently notrelevant to basic stability considerations, they may still play a subtle role in the nonlinear behaviorbeyond the threshold. This is an interesting open question that we have not yet explored.

4.1 Twisted planar ring

We now apply the linear analysis described above to the particular case of the twisted planar ring.We address a classic problem: at what value of the twist, does the planar ring becomes unstableand buckle out of the plane? As we mentioned in Section 2, an analysis based on statics alone doesnot seem to be able to address this problem completely. The time dependent approach answers

this question unambiguously.The stationary solution for a twisted planar ring is given by:

(0) = (k sin(s), k cos(s), ) (17.a)

f(0) = (k sin(s), k cos(s), 0) , (17.b)

where k is the inverse of ring radius (from the center to the central ring axis) and = kT w is,effectively, the twist density (here T w is the total twist in the ring). With this choice of vector ,the director basis (d1, d2) rotates about the central axis with the twist of the rod.

In carrying out the linear analysis some care must be taken with the boundary conditions dueto the periodic structure of the ring: these issues are discussed in (Goriely & Tabor, 1997a). In

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developing the linear variational equations one finds that the and f(1) are non-autonomous.This is readily dealt with by introducing a linear transformation composed of a reflection aboutthe axis d1 and a rotation of angle about d3:

R =

cos(s) sin(s) 0 sin(s) cos(s) 0

0 0 1

. (18)

This transformation is applied to both the basis of the unperturbed system, d(0), and the variables and f(1). This yields the new variables = R. and g = R.f

(1), which together make up thecomponents of the vector (16).

The net result of the stability analysis is a linear system of equations of the form L.x+L.x =0, where

L =

2 ik3 n 2 k3

1 + n2 k2 1 + n2 0 2 ink2

2 0 0 0 n2k2 0k3 1 + n

2

0 2 ik3 n 2 ink2 0 k2 1 + n

2

k2( + n2 1) 2 ik n i k2n 0 1 0ik n n2k2 2 k 1 0 0i k2n 0 n2k2 2 2 0 0 0

.

(19)

Written out explicitly the dispersion relation takes the form:

= 2 k2 n4k2 2 n2k2 + n2 + 1 + k2 n2k2 + 16k4n2

2 k4 + 2 n4 k2 2 k4 + n2 10 n4k4 + 3 k2 3 k4n2 4 n2k2

+4 n6

k4

+ 8 k4

n2

+ + n2

k2

+ 4 n4

k2

+ n6

k4

4

k8n4 (n 1) (1 + n) 2 n4k2 + 2 n4 k2 + 2 n2 4 n2k2 2 22n2 n2k2 + 2 22 + 2 k2 k22

k10 n6 (n 1)2 (1 + n)2 n2k2 22 k2 (20)This relationship, which is an even polynomial of degree 6 in and degree 12 in n, captures theessence of the linear stability problem. Solutions with real positive identify the unstable modeswhich grow exponentially out of infinitesimal perturbations; whereas the other modes, which areeither oscillatory or exponentially damped, will be of the order of the perturbation itself and,therefore, not observable. A typical plot of versus n is shown in Fig. 1. For large n all modesare seen to be oscillatory.

The dispersion relations determine the critical value of the twist T w = k for which thestationary solution first becomes unstable: this is the value of c for which ( = 0) = 0. It is

T wc =

n2 1

. (21)

The first mode to become unstable is the mode n = 2 with twist T wc >

3/. The evolution ofthis mode is shown on Fig. 2 for growing amplitude up to second order in . Our linear analysisconstitutes a direct proof of the twist instability of the ring. Although this condition (21) is wellknown on the basis of static arguments (Zajac, 1962), these arguments cannot prove the actualdynamical instability of the mode after bifurcation. The solution for a given mode can be found

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Figure 1: 2 as a function of n. The total twist is T w = 30 T wc

Figure 2: Evolution of the mode n = 2 for growing amplitude (k = 1/16, = 3/4)

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Figure 3: Linear fundamental unstable solution for the modes n = 3 and n = 5 (k = 1/16, =3/4)

by first finding the real positive value of such that the nth mode is excited (() = 0) andsecond, by computing the null-space of L corresponding to that value of . Having obtained asolution for {, g}, we can obtain = R1. and f(1) = R1.g. The solution X = X(0) + X(1)can then be reconstructed by quadrature:

X(1) =

ds

2d(0)1 1d(0)2

. (22)

The explicit form of X(1) is quite complicated so, instead, we show in Fig. 3 the different modesn = 3 and n = 5. These are the fundamental modes which can become unstable as increases.

The dispersion relations can also help answer another important question: given that a ring

becomes unstable, what is the actual shape that one sees appear? This is the selection problem.Numerical simulations (Klapper & Tabor, 1996; Kleckner, 1996) show that planar rings twistedwell beyond the buckling threshold appear to favor a given geometry as they evolve into thebuckled state. The shape selected by the ring at the onset of instability is determined by theunstable initial modes. Since the instability grows exponentially in time, we expect to see themost unstable mode grow faster than the others and hence to be the first observed. This dominantmode is simply read off the graph of the dispersion relations. By definition, this mode selectionis a dynamical process and cannot be determined from an analysis of the static problem.

We illustrate this by considering the particular case shown in Fig. 1. We see that the modesn = 2 to n = 7 are unstable; however, it is clear that the fastest growing unstable mode is themode n = 4 (The maximum is reached at n 3.855). We therefore expect the mode n = 4 tobe the dominant mode in the dynamics and the ring to select the associated shape. The timeevolution of this mode is shown on Fig. 4. Here, we are only showing the evolution governed by thelinear theory and one should not expect the actual longtime behavior to follow these deformations.However, as mentioned earlier, numerical simulations show that, typically, the growth is dominatedby a single mode and that the early stages of evolution are, in fact, rather well modeled by thelinear theory.

4.2 Twisted straight rod

One of the oldest, and simplest, cases of an elastic instability is that of the straight rod subjectedto twist and tension (or compression) (Love, 1892; Thompson & Champneys, 1996). The classical

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Figure 4: Evolution of the unstable solution for the modes n = 4 for k = 1/16, = 3/4.

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Figure 5: Dispersion relation for P = 3, the straight line corresponds to a trivial straight solution

curve while the parabola is the neutral curve defining the linealry unstable helical mode.

studies are again based on stationary perturbations of static solutions and can only address themost basic stability issues. Here, the full three-dimensional dynamical problem is consideredwithin the framework of Kirchhoff theory.

We consider a straight infinite twisted rod along the x-axis. The twist density in the rod is .The stationary solutions is then given by:

(0) = (0, 0, ) (23.a)

f(0) =

0, 0, P2

. (23.b)

where P2 represents the tension/compression along the rod. The interplay between the parame-ters, P and determine the stability characteristics of the rod. To begin with we only consider a

rod under tension, i.e. f(0)3 > 0; the same analysis applies to a rod under compression.

The linear solutions can be expressed as:

j = et

Axjeins + Axje

ins

j = 1, 2, 3 (24.a)

f(1)j = e

t

Axj+3eins + Axj+3e

ins

j = 1, 2, 3, (24.b)

where = (n) is a solution of the dispersion relation = (, n). The instability threshold(s)is determined by the neutral curves, that is the values of the parameters for which = 0. Thesecurves are given by the solution of (0, n) = 0:

(0, n) = 2 n2 2( 1) P

2

n22

2(

2)2n2 = 0The first instability of the rod can now be determined. The solution of the dispersion relation

is shown as a function ofn for a particular value ofP in Fig. 5. The relation (0, n) = 0 identifies,to first order, two different stationary solutions. The first one, corresponding to the straight lineon Fig. 5, does not correspond to an actual solution (on this curve (1) = (1) = 0 identically).The second solution, corresponding to the parabolic curve in Fig. 5, is physically significant andcorresponds to a helical mode. For this curve the critical values are:

nc =P(2 )

, c = 2 P

. (25)

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Figure 6: Perturbation of a straight rod for nc = 5, c = 8, P = 3. From Left to right A =0, 0.3, 0.6

These new solutions take the form (obtained by the integration of Eq. 13):

X =

s, 2

A

Pcos sP, 2

A

Psin sP

. (26)

Thus for fixed P, the straight rod becomes unstable at the critical twist c and and deforms into

a helix. Some examples of these helical modes are shown on Fig. 6.We conclude by noting that the above discussion has been limited to infinite rods. Thebifurcation condition for rods of finite length exhibits a delay inversely proportional to thesquare of the rod length, i.e. the shorter the rod the sooner the bifurcation occurs. This will bediscussed in the context of the nonlinear analysis described in the next section.

4.3 Helical rod

One of the most fundamental, naturally occurring, filamentary structures is the helix - which ap-pears in fields ranging from molecular biology to magnetohydrodnamics. Furthermore, helical rodsare, subject to appropriate boundary conditions, stationary solutions to the Kirchhoff equations.However, despite their ubiquity, a rather basic question remains unanswered; namely are helical

configurations stable to perturbation?. In fact the results of the previous section on straight rodsmakes this question particularly pertinent. If a twisted straight rod bifurcates into a helix, is thatmode itself stable, or will it undergo a secondary bifurcation into another configuration? Thetools we have developed are able to answer this and other questions about the stability of helicalconfigurations.

Here we consider a helical space curve, xh, parameterized by arc-length, s, turning around acylinder of radius R whose central axis points along the x-axis:

xh = (Ps,R cos(s), R sin(s)) . (27)

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Figure 7: A helical rod characterized by an applied twist , a radius R, and a loop-to-loop distance2P.

where = 1P2+R2 . The choice + defines a right-handed helix for P > 0 (i.e. if you point yourright-hand thumb along the x-axis, your hand will naturally rotate to the right as you move alongthe helix). The height (along the x-axis) per turn of the helix is h = 2P and the length of thecurve per turn is l = 2/. Thus for a helix ofN turns, the total height is H = 2P N and thetotal filament length is L = 2N/. A sketch of a helical rod with an applied axial twist is shownin Fig. 7.

It is straightforward to construct the Frenet triad for this helix beginning with the tangentvector T(s) = xh(s). One obtains

T(s) = (P ,R sin(s), R cos(s)) , (28.a)N(s) = (0, cos(s), sin(s)) , (28.b)B(s) = (R,P sin(s), P cos(s)) . (28.c)

associated The Frenet curvature, F, and torsion, F, are:

F = R2 =

R

P2 + R2(29.a)

F = P 2 =

P

P2 + R2(29.b)

Since we are interested in actual helical rods with axial twist it is convenient to represent thistwist by a ribbon attached to xh which twists around the space curve with a given rotation angle(s). The ribbon construction naturally introduces the director basis:

d(0)1 = cos(s)N sin(s)B, (30.a)

d(0)2 = sin(s)N cos(s)B, (30.b)

d(0)3 = T. (30.c)

which is simply a rotation of the Frenet triad in the plane normal to T.The twist vector may now be computed by substituting the new triad into the twist equa-

tion (1.a) and solving for the unknowns (0). This yields:

(0) = (F sin(s), F cos(s), F + ) . (31)

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where is the twist density of the rod.So far our discussion of the helix has been purely geometrical. To determine the physical forces

acting on our rod - assumed for now to be in a static configuration - we need to solve the timeindependent Kirchhoff equations. Despite the relative simplicity of this calculation, a number ofinteresting results follow. For greatest generality in the constitutive relations (5) we choose theintrinsic curvature components, u to correspond to another helix i.e.

u = (uF sin(us), uF cos(

us), uF + u) . (32)

where uF, uF, R

u, and Pu are, respectively, the Frenet curvature, Frenet torsion, radius, and pitchassociated with this helix and u is a specified twist density. Thus the natural state of our rodis a helix with the above specified parameters. Although we have specified an additional twistdensity u it is easily demonstrated that static solutions for the helix only exist when the stressedand unstressed twist densities are the same, i.e., = u. However, these need not be equal in thespecial limiting case of a helix of zero radius, i.e. a straight rod.

Given (0) and u it is then straightforward to solve the Kirchhoff equations to determine theforce components. These are:

f(0)

= (f0 sin(s), f0 cos(s),

FFf0) (33)

where

f0 = F(F uF) + F(F uF) (34)and we are assuming that F = 0.

To hold a helix in a given shape, it is usually necessary to apply a terminal force, i.e. aforce fx, pointing along the x-axis and applied at the ends of the rod. One may show that fx =f0sec(), where is the pitch angle defined as tan() = R/P. It is clearly possible to constructa free standing helix, namely one for which no terminal forces are required, corresponding tothe condition f0 = 0. Thus, for example, if the unstressed configuration is a naturally straight

rod, (i.e. uF = 0), the free standing condition is = F(1 )/. However, it is importantto note that even if terminal forces are not required, a terminal moment has to be applied tomaintain the helical structure. In the case of the naturally straight rod, this is simply M =(F sin(s), F cos(s), F(1 )).

A linear stability analysis of the stationary helix can now be performed exactly as before;and as in the case of the ring an additional transformation of the form (18) is required to makethe linearized equations autonomous. The actual computations are very long and rely heavily onsymbolic manipulation. The dispersion relations are extremely complicated and are generally notwritten out explicitly. However, given the growth rate for an unstable mode with mode numbern, the explicit form of the excited state takes the general form:

x1(s, t) = P s 2NKR1

nF cos(

ns

N ),

x2(s, t) = R cos(s) K

2 1n N sin(

nNN

s) +2 + 1n + N

sin(n + N

Ns)

,

x3(s, t) = R sin(s) K

2 1nN cos(

nNN

s) 2 + 1n + N

cos(n + N

Ns)

where K = Cet and 1 and 2 are complicated functions of ,n,N,, , F, uF, F,

uF. Their

explicit form is given in (Goriely & Tabor, 1997c).Obviously in considering the stability of a helix there are many choices of parameters: the

geometric parameters, the intrinsic properties, and the elastic constant. For illustrative purposes

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Figure 8: The growth rate 2 solution of the dispersion relation for the (naturally straight) helicalrod as a function of the spatial mode n. The maximum is obtained close to n = 2 (F = 1/8,F = 1/8, N = 5).

we summarize the results for the simplest case: namely a helix constructed from a naturallystraight and untwisted rod, i.e. one for which u = (0, 0, 0) and = 0. A detailed discussion ofthis case as well as the behavior of intrinsically curved and twisted helices is given in (Goriely &Tabor, 1997c). Analysis of the dispersion relation for our chosen problem reveals that there arethree neutral modes: n

N

2= 0,

nN

2= 1 (35.a) n

N

2=

( 2)22F + 2F

2 > 1 (35.b)

The behavior of around threshold can be estimated by a local expansion. Thus around n/N = 1,an expansion in powers of n gives:

2 = (1 )4( nN 1)2 + O

(

n

N 1)4

(36)

Since there is no neutral mode between n/N = 0 and n/N = 1 and locally > 0 around n = 1,we conclude that n > 0 for all modes between n = 1 and n = N

1 and hence that all these

modes are unstable. A typical plot of the dispersion relation is shown on Fig. 8.We are now able to answer a basic selection problem: if a naturally straight rod is maintained

in a helical shape and suddenly released, toward what shape will it evolve? Since all the modesfrom n = 1 to N 1 are unstable and if the helix has more than one turn (i.e. N > 1), the helixwill be unstable. The evolution will be dominated by the fastest growing mode which, for theparameters considered here, can be read off Fig. 8; namely the mode n = 2. The evolving state (atthe level of the linear analysis) is shown in Fig. 9. What is quite striking is that instability tendsto localize the deformation at two points in a way that is characteristic of buckling behavior.

In addition to the case just described we have also studied the behavior of helical filamentswith non-zero intrinsic curvature and find that they are typically dynamically stable. Another

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Figure 9: The time evolution of an unstable (naturally straight) helix. The linear mode n = 2creates two loops (F = 1/8, F = 1/8, N = 5 and K/1000 =0, 2, 4, 6, 8, 10, 12). This linearmode might not represent the true evolution of the nonlinear equations and is a priorionly validfor small values of K (see Section 5)

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situation of interest is to consider the case of free standing helices subject to pulling or pushing.Typically when such a helix is pushed, the resulting instability correspond to a bending of theentire helix, much like the bending of a straight rod. The unstable modes are located around theprincipal neutral mode n = N, so that the instability occur at the scale of the helix itself. Bycontrast when the helix is pulled and twisted, a higher mode nc is excited and the helix deformsthroughout the rod by creating nc > N loops along the helix. This latter case should be contrasted

with the case where the external parameters are held fixed but the intrinsic curvature is changed.Then, the helix undergoes unstable deformations by exciting lower modes nc < N. This leads tothe possibility of buckle formation as shown on Fig. 9. It is this observation that provides themotivation for the dynamical model of buckling described in Section 6.

5 Nonlinear analysis

Although the linear analysis identifies the initial instabilities as a function of the parameters andgives the growth rate of the new solutions, it is limited in many respects. Most importantly, it isonly valid for short times: the exponential growth of the linearized solutions leads to a breakdownof the assumption that these solutions are of order O() and, furthermore, violates our assumption

that the system, which is conservative, is bounded in space and time. Thus as the linear solutiongrows, the nonlinear terms, neglected in the linear approximation, have to be taken into account.

The main idea behind the nonlinear analysis is to develop an asymptotic expansion of thesolution amplitudes, as a function of longer space and time scales, close to the bifurcation point(Newell, 1974). In this regime, the distance from the bifurcation point is of the order of theperturbation itself. This relationship can then be used to introduce new scales on which thearbitrary linear amplitudes can vary. Here we first consider the case of a twisted ring then thestraight twisted rod. In both cases, we take the twist, , as the control (or stress) parameter.

5.1 Nonlinear analysis of the ring

The unstable modes of the ring are discretized due to the boundary conditions and the shape

of the neutral dispersion curves around the origin. Therefore, a nonlinear analysis can only takeinto account the temporal evolution of discrete modes. For twist value greater but close to thecritical twist, there is only one unstable mode, namely n = 2 (See Fig 1 and 2). Thus, we seekto derive an equation describing the temporal evolution of its amplitude A and drop all possibledependence on longer space scales. If the system is close enough to the bifurcation, the differencebetween the twist density and the critical twist c is proportional to the perturbation parameteritself, namely:

2 = c (37)

The new, longer, time scale is t1 = t. The choice of this new scale can be justified by expanding

the dispersion relation in power of (see (Goriely & Tabor, 1997b)). Taking into account thepossibility of the solutions varying on this new (independent) scale, one can now solve the hierarchyof systems obtained by expanding all dependent variables in terms of :

0(0) : E((0); s0, t0) = 0 (38.a)

0(1) : LE((0); s0, t0).

(1) = 0 (38.b)

0(2) : LE((0); s0, t0).

(2) = H2((1)) (38.c)

0(3) : LE((0); s0, t0).

(3) = H3((1), (2)) (38.d)

...

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where Hi is polynomial in its arguments and derivatives in s0, t0, t1.To first order one obtains the linear solution:

(1) = C00 + A(t1)2ei 2sk + c.c. (39)

where c.c. stands for the complex conjugate of the previous expression, 0, 2 C6 are specifiedsuch that 0.0 = 2.2 = 1 and C0 is set such that the ring remains closed when it deforms.

The second-order solution can be found in the same way by solving (38.c):

(2) = 0 + 2ei 2sk + 4e

2i 4sk + c.c., (40)

In order to find a condition on the amplitude A(t1) we demand that the solutions to the third-ordersystem (38.d) remain bounded. Thus, we apply the Fredholm alternative to the system (38.d):

here this consists of integrating H3 against all neutral solutions of the adjoint operator LE:L

01.H3(

(0), (1), (2))ds = 0, (41)

where 1is the adjoint solution to (1)

1 (i.e. L

E.1 = 0) . This compatibility condition gives rise toa differential equation for A as a function of t1, which can be written in terms of t:

2A

t21=

A

27k2 + 8

24

32k3 6(8 11)|A|2

(42)

This equation describes the time evolution of the mode n = 2 after bifurcation. An interestingand important consequence that can be read off directly from the equation is that supercriticalsolutiond ( > c) are always unstable unless > 11/8. In such case, there exist a stationarysolutions with amplitude:

|A|2 =

4

32k3

8 11(43)

Two comments are in order here, first this result is consistent with a result previously obtainedby Maddocks and Rogers (Rogers, 1997) on the basis of the static Kirchhoff equations. Second,this result does not seem very useful since the derivation of the Kirchhoff equations from threedimensional elasticity implies that 2/3 < < 1 < 11/8. However, an argument can be madeto show that rods with noncircular cross sections with high intrinsic twist may actually obey theKirchhoff equations for circular rods but with > 1 (Kehrbaum & Maddocks, 1997). Such systemscould then exhibit behavior like the one outlined here. Moreover, it seems that the experimentalmeasurements of the ratio between bending and twist coefficients for biological molecules such asDNA strands) are also such that 0.7 < < 1.5 (Schlick, 1995). Such high values of might, onthe one hand, justify the applicability of the calculations performed here but, on the other hand,

may raise doubts on the general applicability of the Kirchhoff equations to model biological fibers.We believe that this issue is of fundamental importance and should be further studied.

5.2 The twisted straight rod under tension

We now perform the nonlinear analysis on the twisted straight rod. The main difference here isthat the amplitudes of the nonlinear modes can now vary on longer space scales. Here again, wecan set:

2 = c. (44)

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and introduce the stretched time and space scales:

t0 = t, t1 = t, (45)

s0 = s, s1 = s. (46)

Taking into account the expansion in the bifurcation parameter and the new scales, we now lookfor solutions of the full system order by order in :

0(0) : E((0); s0, t0) = 0 (47.a)

0(1) : LE((0); s0, t0).

(1) = 0 (47.b)

0(2) : LE((0); s0, t0).

(2) = H2((1)) (47.c)

0(3) : LE((0); s0, t0).

(3) = H3((1), (2)) (47.d)

...

where now Hi is polynomial in its arguments and derivatives in s0, s1, t0, t1.To order 0(), the (linear) solution is given by a superposition of the neutral modes; namely

(1)

= X0(s1, t1)0 + Xn(s1, t1)neins0

L

+ X

n(s1, t1)

nei

ns0

L

, (48)

where X0(s1, t1) and Xn(s1, t1) represent, respectively, the slowly varying amplitudes of the axialtwist and the unstable helical mode; n = nc; and

0 = (0, 0, 1, 0, 0, 1) ,

n =

1,i, 0, iP2, P2, 0 . (49)At this order of the functions X0 and Xn are arbitrary and are constant on the scales (s0, t0)but may vary on the longer scales (s1, t1).

In order to derive an amplitude equation describing the evolution of the amplitudes on thesethese new scales we consider the higher-order equations in (47) and look for conditions on the

amplitude to ensure that the solution remains bounded. This condition is again found at O(3

) byapplying the Fredholm alternative. This calculation shows that the amplitudes X0 Y, Xn X,must satisfy the following system of equations:

(P2 + 1

P2)

2X

t21

2X

s21= PX

1 2P|X|2 Y

s1

, (50.a)

2

2Y

t21

2Y

s21= 2P

|X|2s1

. (50.b)

These equations, which are of the form of a system of two coupled nonlinear Klein-Gordonequations, couple the local deformation of the rod X with the twist density Y. We note thecentral role played by the twist density in these equation: if we set Y = 0, it is easy to see that

the stationary solutions may blow up.The space curve can be reconstructed by integrating the tangent vector up to second order in

the perturbation parameter:

x(s, t) =

(1 |X|2)dx

[cos(P x)(Re(X) Y 2Im(X)) sin(P x)(Im(X) Y + 2Re(X))] dx[sin(P x)(Re(X) Y 2Im(X)) cos(P x)(Im(X) Y + 2Re(X))] dx

(51)

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The amplitude equations (50) derive from a Hamiltonian (Goriely & Lega, 1998):

H =

P2

P2 + 1

X

s1

2

+P4

P2 + 1|X|4 P

3

P2 + 1|X|2

Y

1 s1

+

X

t1

2

+

P2

4(P2 + 1)

1Y

s12

+

P2

2(P2 + 1)Y

t12

ds1

such that:

2X

t21= H

X

2Y

t21= H

Y, (52)

where refers to the Frechet derivative.However, equations (50) are probably nonintegrable (they fail the Painleve test (Weiss et al.,

1983) for partial differential equations). Nevertheless, they admit some interesting special solu-tions:

Homogeneous solutions: The spatially independent form of (50) is simply2X

t21=

P3

P2 + 1X

1 2P|X|2 (53)where the twist density decouples from the deformation and is set equal to a constant. Afterintegration, the filament solution is found to correspond to a helix:

x =

s,2X(t)

Psin sP,

2X(t)

Pcos sP

. (54)

Periodic solutions. These solutions are remarkably simple:

X =

Psinns1

L

Y = L2n

sin

2n

Ls1

where = 1 n22PL2 . If we hold the extremities of a finite rod of length L fixed, the constantsKi are determined and we find an envelope solution for the rod:

x(s) =

s,

4L22P 1

2P L

a+a(a cos a+s a+ cos as) ,

4L22P 12P L

a+a

(a sin a+s a+ sin as)

,

where a = P 1/2L. This solution is shown in Fig. 10 for a particular set of parameters.This solution shows a delay in the bifurcation as a function of the rod length: the bifurcationnow occurs at = c +

14PL2 . A gratifying connection between this result and a classical

result is worthy of mention. The stability of a finite straight rod subject to twist and tensionor compression was studied long ago at the level of statics (Love, 1892). For constant twist,

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Figure 10: The stationary solution for (P = 5, = 3/4).

, the static equations are linear and it is possible to show that the bifurcation threshold isgiven by:

2 4P2

2(

1

4L2P2+ 1) (55)

(Here our rod length is scaled by a factor 2 compared to the formula given in (Love, 1892).)Taking the square root of both sides and expanding the square root on the right hand side

to first order, gives

2P

+1

4L2P(56)

This is exactly our delay of bifurcation result deduced from the weakly nonlinear analysis.

Traveling pulses: These solutions can be obtained from the reduction X = u(s1ct1)eit1 ,Y = v(s1 ct1):

X =

P2 + 1

2LP2sech

L

s1 c

2t1

exp

i

k

L

s1 c

2t1

+

L

2t1

Y =

2

(c2 1)P2 + 1

2LP3 tanh

L

s1 c

2 t1

where and c are two free parameters and

2 =2(c2 1)

c2(c2 c20)

(c2 c20) 2c20

, 2 =(c2 c20) 2c20

(c2 c20)2

c20 =2

P2

P2 + 1 =

2P3L2

P2 + 1k =

c

c2 c20which corresponds to a pulse-like solitary wave solutions traveling along the rod with constantspeed c. The possible speed intervals for which a traveling pulse exists are given by thecondition , R. For instance, in the case = 0, we remark that the minimum speed ofthese traveling waves is c2 = /2. This is the speed of the torsional waves obtained fromelementary linear elastic theory(Landau & Lifshitz, 1959). We believe that the solutionsobtained here are the nonlinear version of torsional waves that takes into account the threedimensional structure of the system and allow propagation of waves between regions withdifferent twist densities.

Front: The simplest from of the front can be obtained by setting = 0 in the previousreduction and considering the case where P

2

P2+1 > c2 > /2. One can then find a heteroclinic

connection of the form

X(z) = tanh(z), (57)

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Figure 11: Traveling waves solutions: The pulse (A. c = 4, P = 7 and = 0) and front solutions(B. c = 0.7, P = 1). = 3/4 in both cases.

with K = /( 2c2), 2 = 1/(2P) and 2 = c2P3/ (P2c2 P2 + c2)( 2c2), whichdescribes a front-like solitary waves connecting two different asymptotic states. The twodifferent solutions (pulses and front) are shown on Fig.11.

The amplitude equations (50) are of interest in their own right and have recently been the sub-ject of more theoretical and numerical study (Goriely & Lega, 1998). In particular, the stabilityof the particular solutions has been investigated and a whole new 2-parameter family of partic-ular closed-form solutions has been discovered. This family contains among other, homogeneoussolutions, traveling pulses, fronts and traveling holes (also called gray solitons).

6 Dynamical Model of Buckling

It is easily demonstrated that as a small amount of twist is injected into a straight rod, it deformsinto a helical form (with very small radius) which, as the twist is further increased, tends tolocalize in the middle of the rod and eventually forms a loop. Experimental studies of filamenttwisting (Thompson & Champneys, 1996) have shown that this sequence of bifurcations is quali-tatively correct. Here we draw together the results of the previous sections to develop a dynamicalmodel of buckling in which this process is characterized as a sequence of bifurcations of the solu-tions to the Kirchhoff equations in which the twist density is taken as a control parameter. Thesequence of bifurcations that constitutes our model of spontaneous looping can be summarized asfollows:

Primary Bifurcation

(a) A straight rod, with arbitrary twist, S, and tension P2, is shown to become linearly unstable

at a critical twist value S = 1 (for a given P) and then bifurcate into a helix. The description

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this stage of the process involves a direct application of the linear stability analysis of the straightrod given in Section 4.

(b) The geometry of the helical solution, i.e. its pitch and radius, is completely specified by usingnonlinear analysis to determine its amplitude. Here we are able to use the results of the nonlinearanalysis in Section 6 and the properties of the amplitude equation (50).

(c) The redistribution of the original straight rod twist, S, into the twist, H, and torsion, F,of the new helix is determined using energy considerations. The helix can now be specified by itscurvature F, F, and H.

Secondary Bifurcation

(a) Since the helix itself is an exact solution of the Kirchhoff equations, its linear stability can alsobe analyzed (again by direct application of the results in Section 4) and the critical twist value,H = 2, at which it becomes unstable is determined .

(b) The (linear) post-bifurcation solutions are constructed and a solution with one loop (such asthe one shown on Fig. 12.d), of arbitrary amplitude, B, is identified.

(c) A nonlinear analysis of the unstable helical mode found in step (b) is used to determine theamplitude, B, of the loop. Here a one mode amplitude equation (analogous to the one determinedfor the ring) is obtained of the form:

2B

t21= B(c1 c3|B|2) (58)

where c1 and c3 are extremely complicated algebraic functions of the system parameters whose

explicit form is given in (Goriely & Tabor, 1998a). Since all the parameters involved in the co-efficients of the amplitude equation depend on the control parameter S, the stationary solutionof (58), i.e. B2stat = c1/c3, gives an explicit relationship between the amplitude and the initialtwist density.

Tertiary Bifurcation

(a) A simple criterion to determine the critical B value, Bc, at which the loop will flip is developedfollowing the work of Coyne (Coyne, 1990).

(b) The twist value, 3, at which the loop amplitude equals Bc is found using the properties ofthe solution to (58.

The various special values of the twist, H, 2, 3, can all be related back to the initial twistdensity, S, injected into the straight rod. Thus, in the entire sequence of events S can beconsidered as the control parameter and its critical values at each stage determined explicitly.A sequence of rod configurations as a function of increasing values is shown in Fig.12.

Overall, this multistep dynamical model appears to be consistent with experimental observa-tions, with the possible exception of the very first step. Here we note that in the experimentalwork of Champneys and Thompson (Thompson & Champneys, 1996), the initial helical solution just described (corresponding, in their terminology, to the classic H3 solution), is not actually

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Figure 12: A sequence of helical filaments for varying values of S, a) S < 1, b) 1 < S < 2,c) 2 < S > 3, d) S = 3, e) S > 3. The parameters are = 3/4, P = 1/10, N = 5,1 = 0.2667, 2 = 0, 2684, 3 = 0.27376,and S = 2 + with 103 =0.1, 1.5, 3, 4.9, 5.36, 5.4.

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observed. They find that under rigid loading conditions a helical deformation consisting of onetwist per wave (the state H1) can often appear as a result of continuous deformation i.e. as a pre-bifurcation phenomenon. Rather than observe the three twists per wave helical solution, H3, theysee a localized helical structure L3 (which also has approximately three twists per wave). It is thisstate that then undergoes a writhing instability into the looped state. The main difference betweentheir experimental observations and our theoretical model is that we do not observe the H1 state.

However, this is not too surprising since subsequent studies by Champneys et al. (Champneyset al., 1998) show that the appearance of this solution results from the influence of a small initialcurvature in the rod - an effect not included in our model. Beyond this, their observations andthe rest of our model are consistent with each other.

7 Morphogenesis Driven by Intrinsic Properties

The most general form of the constitutive relationship (5) includes the presence of intrinsic cur-

vatures, (u)1 and

(u)2 , and intrinsic twist,

(u)3 which determine the natural state of a filament.

We now turn to a discussion of the special, if not remarkable, dynamicaleffects that these termscan have in certain biological contexts. The first of these concerns the growth and handedness re-

versal of tendrils in climbing plants, and the second concerns the self-assembly of certain bacterialfilaments.

7.1 Tendril perversion in climbing plants

Tendrils are the tender and flexible side branches of climbing plants, such as the grape-vine, whosecircular meanderings (circumnutation) allows the plant to find a support. On contact with, say, atrellis, the tendrils tissues develop in such a way that it starts to curl and tighten up, and acquirea more robust texture. This curling provides the plant with an elastic springlike connection to thesupport that enables it to withstand high winds and loads. Once the tendril has locked onto thetrellis its total twist content cannot change, since neither the main stem or support can rotate.Thus in order to create a (helical) spring out of a filament with essentially zero total twist, thecoils of the spiral have to reverse at some point so that the tendril goes from a left-handed helixto a right-handed one, the two being separated by a small segment. This handedness reversal istermed perversion and is Natures way of creating a twistless spring. The topic of tendrilperversion fascinated Charles Darwin who discussed it at length in his charming monograph TheMovements and Habits of Climbing Plants (Darwin, 1888). In fact, the topic of perversion inplants has a long history and was noted by many eminent botanists. Indeed Linne himself hasdrawings (Tabula V) of tendrils exhibiting spiral inversion as early as 1751 in his PhilosophiaBotannica.

Tendril perversion is not restricted to plants and is also found to occur in a variety of contextsincluding the false-twist technique in the textile industry (Hearle et al., 1980); the microscopicproperties of vegetal fibers such as cotton ( Textile Institute, 1966); the formation of bacterial

macro-fibers (Tilby, 1977); and in the macrophage scavenger protein, a triple helix with reversedhandedness (Galloway, 1991). However, its most familiar and easily demonstrated occurrenceis in telephone cords: as a coiled telephone cord is first extended and untwisted and then slowlyreleased, a spiral inversion will usually appear in the form of annoying snarls.

A mathematical description of tendril perversion can be provided through a dynamical analysisof the Kirchhoff equations for thin elastic rods in which intrinsic curvature plays a special role.The starting point for the analysis is, again, the linear stability of a straight rod: this time the

rod is ascribed an intrinsic curvature (u)1 and a tension P

2. The linear stability analysis yields

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the dispersion relations:

= n6(n2 + P2)((n2 + P2) K2) n4(n2 + P2)((n2 + 1)(2K2) + 2)2n2(n2 + 1)((n2 + 1) + 4(n2 + P2))4 2(n2 + 1)26 (59)

where for notational simplicity we have set (u)1 K. The neutral curves, determined from the

condition ( = 0, n) = 0, take the rather simple form

(K2 n2) = P2 (60)

Thus for an infinite rod we see that there exists a critical tension, P0 = K/

, such that the rodis unstable for all P < P0. For a finite rod of length L, the first unstable mode is n = 1/L and the

corresponding critical tension is P21 =L2K2L2 . We note that P1 < P0 and that there are a series

of decreasing critical tensions associated with the series of (discrete) modes ending with the lastpossible state which corresponds to the largest integer less than or equal to KL/

.

To second order in the perturbation expansion (see Section 3) the solution corresponding tothe n-th mode takes the form:

x(s, t) =

s X2n sin(2ns) + 2ns2n ,2KX2n P2

2

K2

(1 )cos2

(ns)3P42 7K2P2 + 4K4 ,2Xn sin(ns)n

(61)

where Xn = Re(An). At the level of the linear analysis the amplitude An is still, of course,arbitrary and can only be determined by a nonlinear analysis. Carrying this through at the levelof a single mode analysis (c.f. the discussion of the ring in Section 5) one can determine that theamplitude is:

A2n =43/2(P Pn)(K2 + 3n2)(K2 n2)1/2(12 7)K4 + 2n22(1 2)K2 3n43 . (62)

The solution (61) represents the central, perverted, piece of the problem. We argue that this peri-

odic solution is close to the heteroclinic solution that asymptotically connects helices of oppositehandedness, and using the above results and the standard properties of helices (see Section 4)

we can show that these helices have a Frenet torsion 2F =P2F

K+F(1), and a Frenet curvature,

F, satisfying the condition 2( 1)3F 2K( 2)2F 2FK2 + 2FK = 0. From these cal-culations a rather nice picture tendril perversion emerges. As the tension decreases for a givenintrinsic curvature (or equivalently for an increasing intrinsic curvature and given tension), thetendril reaches a critical state where it looses stability and bifurcates into a solution exhibitingperversion. Beyond threshold this solution asymptotically connects to helical solutions of oppo-site handedness and specific geometry. As shown in Fig. 7.1 this family of helices forms a closedcurve passing through the points (0, 0) (straight line) and (K, 0) (rings). Each point on the curverepresents a helix for given P,K, . The perverted filament asymptotically connects two optimal

helices: a right-handed one (F > 0) to a left-handed one (F < 0)). This figure suggests thatthe perverted filament is, in fact, a heteroclinic orbit in the curvature-torsion plane joining twoasymptotic fixed points (corresponding to the two helices).

7.2 Self-assembly of Bacillus subtilis

The individual cells of the bacterial strainBacillus subtilis are rod-shaped and typically of length3 4m and diameter 0.8m. Under certain circumstances they are found to grow into filamentsconsisting of the cells linked in tandem due to the failure of daughter cells, produced by growth andseptation, to separate (Mendelson, 1990). As they elongate these filaments, which are immersedin a liquid environment (whose temperature and viscosity can be controlled), are observed to twist

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Figure 13: The optimal solution curves in the FF plane together with the two optimal helices(circles) obtained for K = 1/4, = 3/4,n = 1/8 and P = P2 0.286. The inside curve is the oneobtained by the nonlinear analysis for the central piece.

at uniform rate. The degree of twist and handedness can, in fact, be controlled experimentallyand a wide range of states from left-handed to right-handed forms, can be produced. The actualtwist state of the cells seems to be related to properties of the polymers which are inserted intothe cell wall during growth. As they elongate the filaments are observed to whirl and writhe andeventually deform into double-helical structures. These continue to grow and periodic repetitionof this process results in macroscopic fibers (termed macro-fibers) with a specific twist stateand handedness. (A schematic representation of this dynamics is given in Fig. 15.) A strikingfeature of this iterated process is that at every stage of the self-assembly, the handedness of thehelical structures that are created is the same (e.g. a right-handed double helix gives rise to aright-handed four-strand helix and so on). The nature of the environment does, however, influencecertain aspects of the self-assembly. In a viscous environment the basic writhing instability leadsto something of a buckling at the middle of the filament with the formation of a tight centralloop; this is followed by a helical wind-up which starts at the base of this loop. By contrast,in a nonviscous medium, the instability causes the filament to fold over into a large loop closedby contact between the ends of the filament. This closure is then followed by a helical wind-upstarting at that point. In both cases this self-assembly conserves handedness and usually continuesover long periods until macro-fibers, several millimeters long, are formed.

Due its structural complexity, the fact that it grows, and the presence of a fluid environment,

mathematical modeling ofBacillus subtilis dynamics presents many challenges. Nonetheless, someprogress can be made using many of the ideas described above. One fundamental clue to devel-oping a model lies in the observed, handedness preserving, twist-to-writhe conversion. Since thedynamics of this conversion is so fundamental we begin our discussion by first considering thefollowing elementary demonstration involving the twisting of an elastic rubber tube with circularcross section. Held out straight the tube can be said to be a naturally straight rod, i.e. its re-laxed lowest energy state is that of a straight rod without any twist or curvature. The tube is heldin both hands and twisted, say, with right-handed twist. The straight line markers can be seen totake on a left-handed helical path showing that the total twist of the rod is now greater than zero.To return to its natural state the excess twist must be removed. One way to do this is simply to

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Figure 14: Sketch of a helix hand reversal.

let go of one end, which allows the tube to relax by unwinding in the counter-clockwise sense. Thisunwinding takes the form of a twist wave propagating along the tube from the free end towardsthe fixed end. It should be noted that these twist waves correspond to fairly large-scale whirlingor crankshaft like motions in which the tube undergoes quite substantial lateral deformations asit turns. Relief of twist can also be achieved without allowing either end to rotate freely; namelyby letting the restrained ends approach each other and a supercoil form in the counter-clockwise(left-handed) sense. This is the most elementary form of twist-to-writhe conversion. Despite itsconvoluted structure the total twist of the tube when supercoiled is returned almost to zero whichcorresponds to the equilibrium energy state. However, for a fundamental reason, this examplediffers from the situation in Bacillus subtilis, and DNA, where an initial form of a given hand-edness gives rise to a supercoiled structure that is of the same handedness. This fundamentaldifference in behavior is due to the presence of an intrinsic twist in the filament. To illustratethis consider a rubber tube with left-handed intrinsic twist (See Fig. 16,17). The straightenedtube is then twisted at one end in the opposite direction until the markers are approximately

straightened out. The current twist density of the filament in this state would appear to be zerobut its natural state is one of non-zero twist density as indicated by the original helical markerlines. To achieve its natural state the tube must make up for the (apparent) twist deficit. Ifone end is freed the tube winds in a clockwise sense (the same as the helical marker lines) againsending a twist wave down the rod. If the ends are not freed but brought towards each other,the tube will supercoil to relax. In this case the supercoiling is with the same handedness as theintrinsic twist.

For the purpose of modeling the bacterial strands we assume that each individual bacterial cellis a segment of an elastic filament and that it possess an intrinsic twist u3 . Growth of all thecells in a filament uniformly results in an exponential growth accompanied by a gradual reduction

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R

R RR

L

L

L

L

Figure 15: An elementary twist to writhe conversion in Growing Bacillus subtilis

Fixed end Fixed End

Left-handed supercoilingreducing total twist

Or, prevent twisting and bring ends together

Twist wave decreasing the twist

Right-handed twisting

Release one end

Start with a twisted straight rod with zero intrinsic twist density

Fixed end Free end

Fixed end

Figure 16: Twist to writhe conversion with no intrinsic twist

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Right-handed supercoiling

decreasing total twist

twist wave decreasing the twist

Fixed end Fixed End

Or, prevent twisting and bring ends together

Decrease the twist density by left-handed twisting

Release one end

Start with a straight rod with intrinsic twist but zero twist

Fixed end Free end

Fixed end

Figure 17: Twist to writhe conversion with intrinsic twist

of the local twist density. Returning to the rubber tube model described earlier, the intrinsictwist is represented by a helical marker line on the surface of the filament. As the filament growsexponentially the marker line constantly changes pitch as if straightening out along the filamentlength. Changing pitch in this direction away from the initial twist, u3 , results in a filamentthat can be considered undertwisted. The greatest twist deficit, measured by 3 u3 (i. e. thedifference between the local twist density and the intrinsic twist), will be found in the centralportion of the filament whereas the twist deficit will be smallest at the filament ends. This resultsin a large twist gradient at the center of the filament which therefore produces an axial torque.This creates a whirling instability at each of the free ends - in principle these whirling motionspropagate twist back to the center of the filament until the twist deficit has been eliminated.However, two competing processes come into play:

1. the continuous growth of the filament provides an ongoing source of torque at the filamentcenter;

2. due to the growth, the whirling portions of the filament increase in length and experiencean ever increasing drag exerted by the fluid medium.

At a certain point the central, growth produced, torque (one can think of the central portionof the filament as a torque engine fueled by the growth) cannot overcome the viscous drag. Thewhirling motions are effectively blocked and the sort instability scenarios for twisted straight rodsthat we have discussed take over. Now, as the rod continues to grow, the twist deficit builds upand, without the possibility of relief from whirling motions at the ends, undergoes a supercoiling,with the same handedness, to eliminate the twist deficit.

A more quantitative picture has proved to be rather difficult to develop. However, some simplescaling arguments can be proposed. For example, based on crude estimates of balance between thetorque generated by filament growth and the drag experienced by the whirling portions, suggests

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that the critical buckling length might behave as:

l

Ju3Cr

1/3(63)

where is the dynamic viscosity of the fluid medium, r the filament growth rate, the shearmodulus and J the axial moment of the filament, and C an unknown coefficient related to the

drag. The scaling l 1/3 is not intuitively obvious.

Acknowledgments: This work is supported by the Flinn Foundation, NSF grant DMS-9704421and NATO-CRG 97/037.

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Alain Goriely and Michael Tabor- The nonlinear dynamics of filaments