Compressing a rigid filament: Buckling and cyclization

DOI 10.1140/epje/i2007-10230-4

Eur. Phys. J. E 24, 229–241 (2007) THE EUROPEAN

PHYSICAL JOURNAL E

Compressing a rigid filament: Buckling and cyclization

N.-K. Lee1,a, A. Johner2, and S.-C. Hong3

1 Institute of Fundamental Physics, Department of Physics, Sejong University, Seoul 143-743, South Korea2 Institut Charles Sadron, 67083 Strasbourg Cedex, France3 Department of Physics, Korea University, Seoul 136-713, Korea

Received 8 July 2007 and Received in final form 28 September 2007Published online: 9 November 2007 – c© EDP Sciences / Societa Italiana di Fisica / Springer-Verlag 2007

Abstract. We study elastic properties of rigid filaments modeled as stiff chains shorter than their persis-tence length. By rigid filaments we mean that fluctuations around the optimal filament shape are weakand that low-order expansions (quadratic or quartic) in the deviation from the optimal shape are suffi-cient to describe them. Our main interest lies in the profiles of force vs. projected filament length, closureprobability and weakly buckled states. Results may be relevant to experiments on self-assembled biological(microtubules, actin filaments) and synthetic (organo-gelators) filaments, carbon nanotubes and polymersgrafted with strongly repelling side chains, some of which are discussed here.

PACS. 82.37.Rs Single molecule manipulation of proteins and other biological molecules – 87.16.Ka Fil-aments, microtubules, their networks, and supramolecular assemblies – 87.15.La Mechanical properties

1 Introduction

Stiff molecules play an important role in both naturaland synthetic systems [1–3]. In particular, molecules suchas CNT (Carbon NanoTube) and DNA are extremelyuseful from the technological point of view. Other kindsof rigid filament are provided by self-assembling tubulinor actin and synthetic organo-gelators or polypeptides.Over the last decade, the advances in nanotechnology al-low the manipulation of a single molecule [4–7]. Using op-tical/magnetic tweezers, numerous experiments on pullingvarious molecules have been performed. A large numberof theoretical studies [8–11] predict the force-extension re-lation of stiff molecules during pulling which is a centralquantity for many DNA or RNA experiments [4–7,12].However, relatively few studies have been devoted to com-pressed filaments [13–15]. Compressing only makes sensefor rigid filaments that do not coil under thermal fluctu-ations: microtubules (MT), CNT, actin filaments, somesynthetic self-assembled filaments, short DNA fragmentsand short helical or conjugated polymer fragments are afew examples. The common buckling instability of macro-scopic rigid rods also occurs to stiff filaments. Such tran-sitions can be used to design nanoscale switches based onconducting CNT. In a celebrated seminal two-dimensional(2-d) experiment, Dogterom et al. estimated the pushingforce necessary to prohibit growth of microtubule fila-ment [16,17], so-called stall force, from the (stable) buck-led shape of a microtubule filament compressed between a

a e-mail: [email protected]

centrosome which it originates from and a (cell) wall [18].Elasticity of filaments also governs the mechanical re-sponse of systems [19] involving many filaments like net-works built from self-assembling biomolecules (actin) [20],entangled solutions [21] or bundles [13] and assemblies ofbundles [22]. Buckling of a rigid filament and its stronglybuckled shapes are described by standard mechanics in-cluding fluctuation corrections. In this contribution, weestimate the optimal shape and the partition function ofrigid filaments in both weak and strong buckling states.Fluctuations around the straight shape have been consid-ered earlier [13]. We mainly consider filaments with freelyrotating ends which applies, for example, to filaments at-tached to an isotropic bead in a laser trap. We also brieflyconsider filaments with their end orientations imposed,which applies to filaments glued on a rigid substrate (AFMtip) or to a zipping closure of a filament where the relativeorientation for sticking is prescribed. The boundary con-ditions for the filament ends are demonstrated in Figure 1.Schiessel and Kulic argued that the DNA loop conforma-tion on chromatin is mainly determined by energy consid-erations [23] and calculated the minimum energy spectrumfor the size of extra loop using the Euler-Kirchhoff theory.Only planar configurations are sought in this study. Thetheory provided analytic results to some extent. However,for most boundary conditions, the ground-state conforma-tion was obtained only by numerical minimization. Veryrecently the same group also added thermal fluctuationson top of the Euler solution [24] for a filament with im-posed boundary angles.

230 The European Physical Journal E

θ1 θ2θ1 θ =−θ2 1

(a) (b)

Fig. 1. A stiff chain under a compressive force. The verticalpositions of both ends are fixed. The filament adopts its con-formation in accord with boundary conditions: (a) the bothends can rotate freely and (b) the both boundary orientationsare imposed.

Planar configurations are of special interest. Undermost conditions, optimal configurations of filaments areplanar. In some experiments the filaments are first de-posited in a quasi 2-d flow chamber and the system isquasi 2-d by construction. Preliminary adsorption at liq-uid/liquid interface is also used to achieve fast kinetics offilament pairing or closure [25]. Also, real filaments oftenhave non-isotropic section and there is a direction of easybending [26] (cases with spontaneous twist, where the di-rection of easy bending is rotating, are more complicated).In the rest of this paper, we will mainly consider planarconfigurations.

A stiff molecule is modeled as a worm-like chain.Its configuration is described by the curve r(s), the posi-tion in space for a point of curvilinear coordinate s alongthe chain. The tangent unit vector is u(s) ≡ ∂r(s)/∂s.Stiff molecules are characterized by the so-called “persis-tence length”, lp. The angular correlation 〈u(s) · u(s′)〉decays exponentially with |s− s′| over lp along a long fil-ament, due to thermal fluctuations. A filament of totallength S will be called rigid if S ≪ lp. As we focus onthe planar conformation, we leave out the twisting de-formation [27,28] at this point. Later (Sect. 3.4), we willbriefly discuss the influence of twisting on the filamentshape when an out of plane fluctuation is allowed.

The Hamiltonian of the stiff chain associated with thebending energy can be written as

H

kBT=lp2

∫ S

0

ds

(

du(s)

ds

)2

. (1)

This defines the partition function of the stiff chain:

Z =

∫

D(u(s))e−H/kBT (2)

and its free energy F = −kBT lnZ. Some general filamentproperties can be inferred from the structure of the Hamil-tonian. For a given shape, the Hamiltonian H decreaseswith filament length as lp/S. This can be seen by chang-ing variables from s to s/S in the integral in equation (1).For very stiff chains much shorter than the persistencelength for which lp/S −→ ∞, the partition function isdominated by the shape that minimizes H. In these con-ditions, the unconstrained filament is completely straight.Under an external force, the Hamiltonian of the systemconsists of the contributions from the elastic (bending)

energy and the potential energy.

H

kBT=lp2

∫ S

0

dslp2

[

(

du(s)

ds

)2

− f

kBT· (r(S) − r(0))

]

(3)

For a prescribed shape, the Hamiltonian equation (3)under a force is also proportional to lp/S as in the caseof a free chain in equation (1), provided that the force isrescaled by the S-dependent fc ∼ kBT lp/S

2, which will bedefined more rigorously later. This indicates that f/fc, de-termines the optimal shape of the chain. The energy scaleis lp/S in thermal units and the ratio S/lp measures theimportance of fluctuations (some temperature dependenceis formally hidden in lp as lp = κ/kBT with κ the bend-ing modulus, the latter is used in Ref. [13]). Fluctuationsaround the optimal shape are negligible for comparativelyshort filament (vanishing S/lp).

We first review the force-extension relation of the clas-sical Euler-Lagrange solution (Sect. 2). Then in Section 3,we consider the influence of fluctuations and find that theextension is enhanced due to fluctuation at large buck-ling angles. We further consider cyclization triggered bycompressive force. Buckling under various boundary con-ditions is discussed here.

2 The optimal shape of the compressed rigid

filament: Euler elastica

First, we briefly recall the Euler-Kirchhoff theory, that de-scribes the optimal shapes under constraints which serveas reference states for subsequent perturbative treatmentsof thermal fluctuations. The filament can be describedby the contour parameter s and the angle θ(s) betweenits tangent u(s) and a reference direction x. The inte-grations of sin θ(s) and cos θ(s) give y and x coordinatesof the monomer, respectively. We choose the x directionalong the end-to-end vector, so that y(S) = y(0). In theEuler-Lagrange spirit, the Hamiltonian equation (3) cor-responds to the action to be minimized. Under the givenconstraints, the action can be cast as

S =

∫

ds

(

kBTlp2

(

dθ(s)

ds

)2

− µ cos θ(s) − ν sin θ(s)

)

.

(4)

The trajectory of the chain is described by the following“equation of motion”:

kBT lpd2θ(s)

ds2− µ sin θ(s) + ν cos θ(s) = 0, (5)

where µ and ν are the x, y-components of the externalforce f . At a fictitious cut, the torque exerted by the

high-s side on the low-s side is C(s) = kBT lpdθ(s)ds . The

Euler equation expresses the mechanical (possibly unsta-ble) equilibrium under an applied force which states thattorques on a small section of the filament are balanced.

N.-K. Lee et al.: Compressing a rigid filament: Buckling and cyclization 231

θ0

θ0

x

y y

x0 0

s=S/2s=−S/2s=−S/2 s=S/2

a) b)

Fig. 2. (a) Coordinates of Euler elastica for symmetric con-formations with clamped boundaries. (b) The excited state atgiven θ0.

The Euler equation (Eq. (5)) yields the following first in-tegral:

1

2kBT lp

(

dθ

ds

)2

+ µ cos(θ(s)) + ν sin(θ(s)) = c. (6)

For the filament with freely orienting ends, there isno external torque exerted on the ends hence C(0) =C(S) = 0. The torque balance on the whole filament im-poses that the force is along x, i.e. µ = −f and ν = 0,as can also be seen by integrating equation (5) over θalong the filament. The Hamiltonian invariant in equa-tion (6) then imposes cos(θ(0)) = cos(θ(S)). It is easy tosee that the optimal shape under compression correspondsto θ(S) = −θ0 where we set θ(0) = θ0. Optimal configura-tions under compression are symmetric for freely orientingends. The first integral equation (6) is simplified to

dθ

2ds/S= π

√

f

fcsin(θ0/2)

(

1 − sin2(θ/2)

sin2(θ0/2)

)1/2

, (7)

where fc = π2kBT lp/S2 will turn out to be the critical

force for buckling (here thermal fluctuations are ignored)and f is the magnitude of the compression force. It is nownatural to take the origin of the curvilinear coordinate sand of the spacial coordinates (x, y) at the middle of thefilament (see Fig. 2). A successive integration of equa-tion (7) leads to the optimal filament orientation θ(s):

sin(θ(s)/2) = | sin(θ0/2)| sn(

πs

S

√

f

fc, sin2(θ0/2)

)

,

cos(θ(s)/2) = dn

(

πs

S

√

f

fc, sin2(θ0/2)

)

,

(8)

and the filament shape

x(s) = − s+2

πS

√

fc

fE

(

πs

S

√

f

fc, sin2(θ0/2)

)

,

y(s) = 2 sin(θ0/2)S

π

√

fc

f

×(

1 − cn

(

πs

S

√

f

fc, sin2(θ0/2)

))

,

(9)

where sn(u,m), cn(u,m) and dn(u,m) are Jacobian ellip-tic functions and E(u,m) is an incomplete Elliptic func-

tion [29]. From the first of equations (8), we see that π2

√

ffc

0 1 2 3f / f

c

0

1

2

3

4

θ0

(b) (a)

(c)

Fig. 3. Buckling angle θ0 below/above the critical force fc

under the condition of free rotation at boundaries. (a) Euler-Lagrange solution given by equation (10), (b) asymptotic be-

havior close to the transition θ0 =q

8 f−fc

fc(c) the mean values

of the buckling angle in the first-mode approximation.

0.0 1.0 2.0 3.0f/f

c

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x/S

X/S Euler-Lagrange

(S/lp )/[2π2(f/f

c-1)]

<δx>/S Gaussian

<δ2x>/S

2

(X+<δx>)/S Gaussian

(X+<δx>)/S quartic

lp/S=3

Fig. 4. Extension vs. force relations of a filament withlp/S = 3. Bold line is from the ground-state description of theEuler-Lagrange equation. The correction to the ground state isevaluated when fluctuations are taken into account before andafter the buckling transition in quadratic and quartic order ofδθ. The size of fluctuations 〈δx〉 and 〈δ2x〉 are also indicatedby symbols.

is the quarter period given by the complete elliptic integralK(m). This determines the force-buckling angle relation-ship:

√

f

fc=

2

πK(

sin2(θ0/2))

. (10)

The above equations describe the buckled state. It is clearfrom equations (9) and (10) that the buckling angle θ0vanishes at the finite compression force fc which was in-deed announced as the critical force. The two branchesand the standard square-root expansion close to the crit-

ical force θ0 =√

8√

f−fc

fcare recalled in Figure 3. From


-0.6-0.4-0.2 0.2 0.4 0.6

-0.4

-0.2

0.2

0.4

-0.6-0.4-0.2 0.2 0.4 0.6

-0.4

-0.2

0.2

0.4

-0.6-0.4-0.2 0.2 0.4 0.6

-0.4

-0.2

0.2

0.4

-0.6-0.4-0.2 0.2 0.4 0.6

-0.4

-0.2

0.2

0.4

-0.6-0.4-0.2 0.2 0.4 0.6

-0.4

-0.2

0.2

0.4

-0.6-0.4-0.2 0.2 0.4 0.6

-0.4

-0.2

0.2

0.4 f/fc = 1.39f/fc = 1.02

f/fc = 2.19 f/fc = 2.69 f/fc = 12.52

f<fc

θο = 0 θο = π/6 θο = π/2

θο = 2.28 θο = 2.5 θο = π/2

(a) (b)

(e)(d)

(c)

(f)

Fig. 5. Various conformations of a stiff chain below and above the bifurcation transition obtained from the Euler-Lagrangeequation.

equations (9) the end-to-end extension of the filament is

X = S

(

−1 + 2E(

sin2(θ0/2))

K(

sin2(θ0/2))

)

(11)

and later this will be corrected by fluctuations. Force ex-tension laws are represented in Figure 4. The closed con-figuration corresponds to f/fc = 2.19 and θ0 = 2.28. Thisshape is of special interest as it has the lowest bendingenergy among all possible closed filament shapes (up tocorrections due to fluctuations). The angle between inter-nal filament strands at contact is 1.42.

Equation (10) is not the only solution to equation (7)but the one with the lowest energy. There are also excitedstates representing local minima of the energy. These are

given by√

ffc

= p 2π K(sin2(θ0/2)) with p an odd integer.

For a given θ0, all optimal states (including the groundstate p = 1) correspond to the same end-to-end extension(as can be also seen from E(pK) = pE(K)). Comparing thebending energies, Ub = f(X − S cos(θ0)), of the variousstates for the same separation X, Ub(p) = p2Ub(1), thefirst excited state has a bending energy nine times thatof the ground state. In microtubule networks [30], due tothe presence of surrounding cytoskeletons, the observedbuckling wavelength is also much smaller than the fila-ment length. Figure 5 displays the shapes of the filament,so-called Euler elastica, for some typical buckled states.Figures 5(c) and (f) illustrate the ground state and firstexcited state for θ0 = π/2. The total energy Ut = Ub +Xfcan be expressed in terms of force and m = sin2(θ0/2):

Ut(f,m)

kBT= Sf

(

2m− 3 +4E(m)

π

√

fc

f

)

. (12)

3 Fluctuations

Next we will take into account fluctuations around the op-timal shape θEL(s). To do so, we may expand the Hamil-tonian around θEL(s), H = HEL + Hfl. Below we find

it more convenient to choose the origin of s at a filamentend. To Gaussian order the fluctuation Hamiltonian Hfl

reads

Hfl

kBT=

∫ S

0

ds

[

lp2

(

dδθ(s)

ds

)2

+µ

2kBTcos θEL(s)δθ(s)2

+ν

2kBTsin θEL(s)δθ(s)2

]

, (13)

where δθ(s) = θ(s) − θEL(s) is the deviation from theEuler-Lagrange solution. The bending contribution can be

transformed to[

lp2 δθ(s)

dδθ(s)ds

]S

0− lp

2

∫ S

0dsδθ(s)d2δθ(s)

ds2 by

integration by part. For both fixed boundary angle andfreely rotating boundary angle, the bracket vanishes.

Defining the operator H = −lp d2

ds2 + U(s) with U(s) =µ

kBT cos θEL(s) + νkBT sin θEL(s), we arrive at

Hfl

kBT =12 〈δθ|H|δθ〉. The partition function associated with Hfl

can be calculated from the spectrum of H. In the stan-dard situation, the expansion is carried out around a sta-ble equilibrium and the spectrum is positive.

Zfl =∏

n

√

2π/λn, (14)

where λn’s are the eigenvalues of H. This product will turnout to be divergent due to short-wavelength modes. Forreal filaments, the finite thickness is a natural cut-off, asshorter modes cannot be treated in our description. In theremainder, we rather normalize Zfl by its value in theabsence of force, Z0

fl, to eliminate this divergence. Undera fixed compression force as considered here, there is anunstable mode, corresponding to a negative eigenvalue.The physical reason for this mode is clear: the compressionequilibrium we are describing is unstable and an extraconstraint is necessary to prevent, say, the global rotationtowards the stable equilibrium where the applied force ispulling and the Euler-Lagrange shape is straight. This willbe discussed in the following sections.


3.1 Fluctuation in unbuckled states

A slightly constrained filament under f < fc does notbuckle and its optimal shape is straight (θEL(s) = 0).Fluctuations do depend on the applied force: we expect acompression force to enhance fluctuations, much so if thebuckling transition is approached. An extensional forceirons the filament and the projected length ultimatelymatches S for a diverging force. Let us take the exampleof filaments with freely orienting ends (no localized torque

is applied to the filament edges). The eigenmodes of H

are ψn(s) =√

2S cos(πns/S) for n > 0 and ψ0 = 1/

√S

with the spectrum λn = n2fc/kBT − f/kBT . The moden = 0 is unstable which corresponds to a global rota-tion of a straight filament. The stability is marginal for avanishing force where the system is invariant under rota-tion. A given deviation from the straight shape is repre-sented as δθ(s) =

∑

nAnψn(s). The global displacement

y(S)−y(0) =∫ S

0δθ(s)ds = A0

√S only depends on A0, to

lowest order in δθ. It is hence enough to constrain the endsto be aligned on x to remove the rotation instability, with-out affecting the other modes. In practice one could usean anisotropic laser trap. It is easy to see how a harmonicy-potential can stabilize the n = 0 mode1. Assuming astrong reduction of this mode, for simplicity, we omit thecorresponding factor in the infinite product:

Zfl

Z0fl

=

√

√

√

√

√

π√

ffc

sin(

π√

ffc

) , (15)

Ffl −F0fl =

kBT

2log

sin(

π√

ffc

)

π√

ffc

, (16)

which characterizes the decrease of the free energy by thefluctuations. For weak compressive forces much weakerthan the one causing the buckling transition Ffl −F0

fl ≈−kBTπ2

12ffc

, this leads to the length stored in fluctuations

at a vanishing force ∆L = S2

12lp, which is small compared

to S in the limit of rigid filament (S/lp ≪ 1). As expected,the free-energy reduction diverges at the buckling insta-bility. The length stored in fluctuations,

∆L =π2kBT

4fc

cot(

π√

ffc

)

π√

ffc

− 1

π2 ffc

, (17)

becomes unphysical (formally larger than S) for (fc −f)/fc < S/(π2lp) (it was argued that this criterion qual-itatively locates the buckling transition under fluctua-tions [13]) and formally diverges at the transition as ∆L ≃

1 Assume a harmonic y potential Vy = 1

2Ky(δy)2 acting on

one end (bead) the other end being fixed in space and freelyarticulated. This contributes an extra energy 1

2KyS(A0)

2 re-

moving the instability provided Ky > fS. The omited factor in

equation (16) is 1/p

1 − f/(KyS) ≈ 1.

− kBT2(f−fc)

when calculated to quadratic order, this we fur-

ther discuss below. Let us first notice that equation (16)remains true under an extensional force of the magnitudef , provided that “sin” is replaced by “sinh”. In this way,one can calculate the length stored in fluctuations underextension and recover that it vanishes as 1/

√f at high f ,

∆L = S4

√

kBTflp

(if two equivalent bending directions are

allowed (3-d) the length stored in fluctuations is doubled).The latter result, was obtained earlier [8] for the strongstretching regime of a long semi-flexible chain and hasbeen proved very useful for DNA stretching experiments.

3.2 Buckling

Close to the buckling transition, the expansion of theHamiltonian equation (13) in the An has to be carriedout to higher orders. We expect the main physics to becaptured by expanding to quartic order in the amplitudeA1 of the unstable mode and to quadratic order for others:

Hfl

kBT≈∫ S

0

{

− lp2δθ

d2δθ

ds2− f

2kBT(δθ)2+

f

24kBT(δθ)4

}

ds

=λ1

2A2

1 +

∞∑

n=2

λn

2A2

n

+f/S

24kBT

{

3

2A4

1 + 2A31A3 +

∞∑

n=2

6A21A

2n

}

. (18)

After having integrated out the amplitudes of the sta-ble modes, we arrive at the reduced partition functionZfl/Z0

fl:

Zfl

Z0fl

=

√

fc

2πkBT

∫ +∞

−∞

dA1 exp

[

−H1

kBT

]

,

H1 =H1+1

2log

sinπ

√

ffc

(

1−A2

1

2S

)

π

√

ffc

(

1−A2

1

2S

)(

1− ffc

(

1−A2

1

2S

))

, (19)

where H1 collects terms in equation (18) depending on A1

only: H1 =π2lp

S (A2

1

2S (1 − f/fc) + 14

ffc

(A2

1

2S )2). No further

approximation was made during the integration over theamplitude of the stable modes. The above expression equa-tion (19) of H1 makes sense, provided the argument of thelogarithm is positive. For A2

1/(2S) < 1, this requires f <4fc, for A2

1/(2S) > 1 the expression still formally makessense but “sin” is better rewritten as a “sinh”, yielding:

12 log

sinh π

r

ffc

(A2

1

2S−1)

π

r

ffc

(A2

1

2S−1)(1+ f

fc(

A21

2S−1))

. It should however be noted

that largeA2

1

2S implies large fluctuation of the buckling an-gle. It means that the initial small buckling angle expan-sion becomes inaccurate. Assuming that the fluctuation ofthe buckling angle is smaller than one (which does apply


at least away from the buckling transition) we may expand

the effective Hamiltonian H1 to first order in A21/(2S):

H1 = . . .+H1 +

(

3

4+

1

2(f/fc − 1)

−π√

f/fc

4cot[

π√

f/fc

]

)

A21

2S+O

(

(

A21

2S

)2)

, (20)

where . . . stands for the contribution independent ofA1 summing the stable modes in the Gaussian approx-imation. If we further expand the right-hand side ofequation (20) for f/fc close to unity,

H1 = . . .+H1 +3

8

A21

2S+O

(

(

A21

2S

)2)

. (21)

The latter expression is also obtained from a direct ex-pansion early in the calculation, it amounts to replacingthe powers in the amplitudes of stable modes in theHamiltonian equation (18) by their thermal averages. Inthe following we use the expanded equation (21) which isaccurate, provided the fluctuation of the buckling angleis smaller than unity:

Z1 =

∫ +∞

−∞

dA1 exp

[

−1

2

(

fc − f)

kBTA2

1 − Γ4A41

]

. (22)

The integration over An other than A1 renormalizesthe critical force (second-order vertex) by corrections oforder of S/lp. A similar correction is obtained for thefourth-order vertex. In our calculation this however doesnot make Γ4 negative and the transition would remaincontinuous. We find fc = fc(1 + 3

8π2

Slp

). Our calculation

holds for buckling angles smaller than unity. It fails toprecisely locate the critical force and to describe the verycritical region if fluctuations of the buckling angle areof order unity [13]. Keeping this limitation in mind, wecalculate the fluctuations of the buckling angle and theforce extension curve from equation (22). Close to the

transition (f ∼ fc), the fluctuations are large but finite,and dominated by the mode n = 1. Hence 〈θ20〉 = 2

S 〈A21〉.

From equation (22), at f = fc,

⟨

θ20⟩

fc=

2

π

Γ [3/4]

Γ [5/4]

√

S

lp. (23)

When numerical estimates of the Euler Gamma functionΓ [x] are inserted, the amplitude is close to 0.86. Formallythese fluctuations remain weak, but get much largerthan those in the states far from the transition wherethey are proportional to S

lp. The partition function Z1

equation (22) can be integrated both below and above fc

0.6 0.8 1.2

2

4

6

8

10

12

f/fc

Asymptotics

EulerLagrange Solution

02<θ >/<θ >

c02

1 . 0

Fig. 6. Reduced buckling angle fluctuation〈θ2

0〉

〈θ2

0〉c

(see text).

The average buckling angle from the first mode approximationis shown together with its asymptotic close to the transition(dashed line). We choose lp/S = 20.

as a function of ǫ = 2 fc−f

fc

√

fc2S

kBTf :

Z1(ǫ) = 2

(

kBTS

f

)1/4

g−,+(ǫ),

g−(x) =1

2

√xK1/4

(

x2/8)

exp(

x2/8)

, (24)

g+(x) =π

23/2

√−x(

I1/4

(

x2/8)

+ I−1/4

(

x2/8))

× exp(

x2/8)

,

where g−,+ expressed in terms of the modified Bessel func-tions Kα[x] and Iα[x] apply below and above the transi-tion, respectively. Far below the transition where g−(x) ∼√

πx , we recover the result of the quadratic approximation

up to the replacement of fc with fc which is unimportantin this limit. At fc, g−(0) = g+(0) = Γ (1/4)/2. The fluc-tuation of the buckling angle can be derived from thepartition function. Its value 〈θ20〉 normalized by that atthe transition 〈θ20〉c is shown in Figure 6. For convenience,only the first mode that dominates close to and above thetransition is taken into account. In the same coordinates,the Euler-Lagrange solution 2A2

1/S = 8(1 − fc/f) in thefirst mode approximation together with its asymptoticbehavior close to the transition (already given earlier) isrepresented. It can be seen that close to the transition,the buckling angle fluctuates widely over both branches ofthe bifurcation. For f

fc≫ 1, the fluctuations approach the

value of the Euler-Lagrange solution. This means that thefilament is localized close to one of the branches and hasonly a small probability to be found with an intermediatebuckling angle. In practice, the filament can only switchbranches upon crossing a huge energy barrier and for aphysical waiting time, it appears localized on one branch.Next we briefly discuss fluctuations around one branchfor a buckled filament away from the transition.


3.3 Fluctuation around one branch, a buckled state

Not so far from the transition (for f−fc

fc< 1) the one-

mode description only with the amplitude A1 providessufficient accuracy. The Euler-Lagrange branch is then

approximated by A21 = f−fc

4kBTΓ4

and fluctuations around

the upper branch A+1 are governed by the Hamiltonian

δH1 = f−fc

kBT (δA1)2 + 4Γ4A

+1 (δA1)

3 +Γ4(δA1)4, where the

cubic term indicates a bias of the fluctuations towards theopposite branch. This free energy is double-well shapedwith two equivalent minima located on the branches sep-

arated by a barrier of height (f−fc)2S

fkBT . As expected the

barrier becomes relevant if the buckling angle markedlyexceeds the fluctuation estimated in equation (23). Underthese conditions, the fluctuations around one branch be-come relevant and 〈(δθ)2〉 = fcS

(f−fc)π2lp. The partition

function Z1 ∝√

πf−fc

determines the variation of the pro-

jected length 〈δx〉S = fcS

2π2(f−fc)lpand its fluctuation 〈(δx−

〈δx〉)2〉 = 2〈δx〉2. The expression of 〈δx〉 obtained asymp-totically below the transition (where 〈δx〉 = −∆L) is iden-tical, indicating that the force-extension curve rounded byfluctuations cuts the Euler-Lagrange curve in the transi-tion region. Such a cut is provided by the quartic approx-imation discussed above. This seems to be a typical 2-deffect also found in simulations with fixed boundary anglesin [24] based on bead-spring model.

In the general case we may seek for the spectrumof H for a finite buckling angle θ0 6= 0. H has off-diagonal elements 〈ψn|H|ψm〉 when expressed in the base

ψn =√

2/S cos(πns/S),

〈ψm|H|ψn〉 = n2 fc

f− 2

π

√

fc

f

∫ π/2√

f/fc

−π/2√

f/fc

duψm(u)

×(

1 − 2 sin2(θ0/2) sn2(

u, sin2(θ0/2)))

ψn(u)

= hmnkBT

f(25)

with u = πs/S√

f/fc. Diagonalization gives eigenvaluesλn of buckled states. The product of positive eigenvaluesreduced by the zero force values λn/λ

0n converges when

higher and higher modes n are included thereby increasingthe size of the H matrix. In practice, it is usually sufficientto consider a 20× 20 matrix. The force-extension relationcorrected by fluctuation (X + 〈δx〉) is shown in Figure 4.

With cyclization kinetics such as protein-mediatedDNA looping [31,32] in mind, the closed (or nearly closed)conformations are of special interest [33]. From the Euler-Lagrange equation, we find that the stiff filament is closedat force fcyc/fc = 2.18 (see Fig. 5d) with a bending en-

ergy Eb = 1.42π2 lpS kBT . The case when lp/S remains

moderate, say no more than 5, is of most practical in-terest, here again fluctuations become relevant. The shift〈δx〉 = δF/δf in x away from closure is 0.055S2/lp. As aconsequence, closure (〈x〉 = 0) corresponds to a some-

what larger force shifted by δf = kBT〈δx〉σx

with respect

to the Euler-Lagrange estimate with σx = −kBTd〈x〉/dfwhich is the fluctuation under force fcyc. This shift in forcecorresponds to an additional shift in free energy δF =

kBT(〈δx〉)2

2σx. At this point we need to estimate σx, from the

Euler-Lagrange solution σx = 0.35 S3

π2lp(the fluctuations

add a correction ∝ S4/l2p), this is typically larger than theshift away from closure at fc. The reduction of partitionfunction when imposing 〈x〉 = 0 as compared to the free

state is exp −(Eb+δF )kBT with δF/kBT = −0.32 + 0.04S/lp.

The (positive) correction arising from the shift 〈δx〉 issmall for lp/S > 1.

3.4 Symmetric fixed-angle boundary condition and azipped state

So far, we have focused on the special case in which thereis no torque exerted on the filament ends. The resultingconformations are naturally symmetric. In this section, weconsider a symmetric conformation with the end orienta-tions imposed. Quite often binding of two filament strandsrequires a special relative orientation [31,34,35]. Formallythe boundary condition requires δθ(0) = δθ(S) = 0. Whenboth angles are fixed to θ0 (up to sign) to impose asymmetric configuration, there is a circular equilibriumstate (f = 0) with X = S sin θ0

θ0

. Smaller extensions corre-spond to squeezing. When the pushing force is increased,a state is reached where imposing the chosen angle doesnot require any external torque. This state is describedby equations (10, 11) for freely rotating boundary anglediscussed above. At the force given by equation (10), theshape displays an inflexion point at the filament edges.For a higher force, the inflexion points move inwards (seeFig. 7(a)). The parameter m of the Jacobian Elliptic func-tions is linked to the angle θc at the inflexion point:m = sin2(θc/2). The angle at the inflexion point θc and theboundary angle θ0 are linked by equation (8) at s = ±S/2:

cos(θ0/2) = dn

(

π

2

√

f

fc, sin2

(

θc

2

)

)

. (26)

If the filament is free to twist out of the plane, it will re-lease some constraint by building a loop if θ0 < π/2 andthe angle then switches from θ0 to π − θ0 (i.e. Figs. 7(a),(b)). When the constraint is increased, the first possibletwisted shape has an inflexion point at the edges and hencecorresponds to the parameterm = cos2(θ0/2). If the initialforce is further imposed, the end-to-end distance increasesbeyond the neutral point X = S sin θ0

π−θ0

to reach a new equi-

librium. This is illustrated in Figure 7(c) for θ0 = π/6.Any cost for twisting will suppress or delay the transi-tion. We do not want to discuss twisting energy in detailhere as it depends on boundary conditions imposed forthe torsion angle φ. If angle φ is free (as could be assumedif the end is inserted into a hollow cylinder fixing θ butallowing for free rotation φ around the fibre axis), there isno penalty for twisting. If the end sections are constraintwith a fixed orientation φ, twisting requires a high torsion


cθ0θ =0θ

cθ

0π−θ

(a) (b) (c)

Fig. 7. Fixed-boundary-angle condition with θ0 = π/6 before and after twisting with f/fc = 2.217 (a) At a large compressiveforce, the inflexion points exist at internal positions, θc > θ0 and m = sin2(θc/2) = 0.93. (b) Right after twisting, the inflexionpoints are at edges θc = θ0. Here m = sin2(θ0/2) = 0.57 (c) The corresponding pulling force leads to the stretched conformation.The inflexion points exist on external points where contour shapes are extended. θc < θ0, m = 113.5.

-0.2 -0.1 0.1 0.2

0.1

0.2

0.3

0.4

0.5

-0.2 -0.1 0.1 0.2

0.1

0.2

0.3

0.4

0.5

θ = 2/3π0 θ =π/20

0 2 4 6 8f/f

c

-0.5

0

0.5

1

X/S

a b

a b

Fig. 8. Force-extension relations when the end orientation θ0 is fixed as π/6, π/3, π/2, 2π/3 and 5π/6 from top to bottom.(Twisting is not allowed.) Closed shapes with preferred relative orientations are illustrated: (a) θ0 = 2π/3 with m = 0.785 andf/fc = 2.56 and (b) θ0 = π/2 with m = 0.731 and f/fc = 4.01, where the loop ends are antiparallel, leading to zipping.

Table 1. Bending energies and fluctuation of closed shapeswith the imposed end orientations. The numbers in parenthesis(γ = 2θ0 − π) are the angles between the strands at closure.

θ0(γ) fx/fc Eb/(π2lp/S)(kBT ) m δ F (kBT )π2(0) 4.01 1.85 0.731 2.869

2π3

(π3) 2.56 1.44 0.785 2.806

2.28(1.42) 2.20 1.43 0.825 2.7923π4

(π2) 2.0 1.43 0.858 2.788

4π5

( 3π5

) 1.6 1.47 0.958 2.780

energy Et = 2π2kBT lt/S with lt a characteristic lengthwhich is typically larger then lp (by about a factor 2 forDNA). Any soft constraint on φ like an elastic constraintmodeled by a harmonic potential is in between. If the en-ergy for twisting is not negligible, the twisted shape is pro-hibited or becomes stable at shorter end-to-end distances.

Together with equation (26), the force-extension re-lation at a given terminal orientation can be computed

from equation (9). X = 2πS√

fc

f E(π2

√

ffc, sin2(θc/2))− S,

wherem = sin2(θc/2) replaces sin2(θ0/2) in equation (26).In Figure 8, we demonstrate computed force-extension re-lations for specific end orientations. In particular, someclosed shapes (X = 0) are illustrated for specific θ0. Wecompute compressive force fx acting on the loop termini

and the corresponding bending energies in Table 1. Theshift in extension due to fluctuation 〈δx〉 have positivevalues at all imposed angles. Thus, the required force toclose a filament is larger than the classical estimation dueto the fluctuations.

Table 1 demonstrates the bending energies and thefluctuations2 of closed shapes for given imposed bound-ary angles θ0. The numbers in parenthesis (γ = 2θ0 − π)represent the angles between the strands at closure. Thebending energy minimum is found at θ0 = 2.28, γ = 1.42,nothing but the closure conformation with freely rotatingends (see Fig. 5(d)). On the other hand, the free-energycost due to reduction of fluctuations (with respect to free

2 When the boundary angles are fixed, the shape fluctua-tions are represented by linear combination of basis functionsvanishing at the boundary δθ(S) = δθ(0) = 0. We thus expand

δθ(s) =P

n Bnξn with ξn =p

2/S sin(πns/S). ξn satisfies anequation similar to equation (25) where θ0 in equation (25) isnow replaced by θc given by equation (26):

〈ξm|H|ξn〉 = n2 fc

f− 2

π

s

fc

f

Z π/2

√f/fc

−π/2

√f/fc

duξm(u)

×`

1 − 2 sin2(θc/2) sn2(u, sin2(θc/2))´

ξn(u)

= hmnkBT

f.


filament) δF is largest when the strands are antiparallelat closure. δF is independent of the ratio lp/S while thebending energy increase with lp/S. Thus the favorable clo-sure angle shifts (in practice slightly) from freely rotatingcase γ = 1.42 to larger closure angle γ as chain flexibilityincreases.

Zipped state. For rigid filaments small loops are dis-favored and we may focus on the zipping cyclization.In our ideal (planar) filament picture, a natural require-ment is that strands are antiparallel. Figure 8(b) corre-sponds to an interesting case where θ0 = π/2. All shapesfor f/fc > 1.39 (m > 1/2) display internal points withangle π/2. There is an internal strand satisfying all con-straints and its shape provides a solution in the small forceregime (after a proper size rescaling). A closed shape is ob-tained for m = 0.731 for a force of f/fc = 4.01. The bend-

ing energy of the closed shape is Eb = 1.85π2 lpS kBT , where

the prefactor is slightly higher than for the case of freelyorienting filament edges. Note that bending the filamentinto a circle would correspond to higher bending energywith prefactor 2 instead of 1.85. If some zipping closure isinduced by filament/filament binding with an energy gainδb per persistence length, the free energy of loop size s is

F = δb

2lps+1.85π2 lp

s in kBT units. By minimizing the free

energy with respect to s, we find the terminal loop sizest = lp

√

3.7π2/δb, which can be quite smaller than lp fora sufficiently large binding energy δb. Again fluctuationcorrections are generally relevant.

Similarly as described in Section 3.3, we find the shiftin δx away from the closed state to be 0.058S2/lp. Thisshift is about similar (slightly larger) to that under thefreely rotating boundary condition.

3.5 Mixed boundary conditions

In this section, we discuss experimental situations wherethe boundary conditions are not symmetric.

Often, the filaments are subjected to mixed boundaryconditions with one edge freely orienting and the otherwith a constrained orientation. The mixed boundary con-ditions are relevant to several experiments. In experimentson microtubules by Dogterom et al. [16,17] one end of themicrotubule is blocked by a wall roughness and can rotatefreely while at the other end the orientation β is adjustedto zero. Another example can be found in recent experi-ments of the deposition of CNT filaments in a two dimen-sional (2-d) chamber. CNTs are deposited on a stripedsurface [36] in a quasi 2-d flow chamber. As the filamentsare rather rigid, they lie more or less straight on the ad-sorbing stripes. One edge typically fixes first. There areinteresting stripe-edge effects shown in Figure 3 of refer-ence [36]. If the adsorbing stripe width is smaller or com-parable to the filament length, a filament typically leavesthe stripe under an angle β and eventually binds back intothe stripe. Prior to re-adsorption, the free end experiencesno external torque and orients freely. By symmetry thepreferred shape back into the stripe is symmetric. In or-der to minimize the bending energy, the turn involves the

xf

0

x

o φ

β

x’

λ

S

Sθ

τFig. 9. The coordinates of Euler elastica in mixed boundarycondition. The inflexion point is chosen as the origin. The ac-tual shape is obtained by rotation of φ from the symmetricshape defined between two inflexion points. θ0 is the angle atinflexion point and β the imposed angle at s = S.

whole strand length available. However, as the adsorbedfilament cannot reorient on the substrate but only slideat a fixed local orientation (as seems to be the case), theshape subsequently tightens under a longitudinal pullingforce equal to the binding energy per unit length. The finalorientation recollects the landing angle of the freely rotat-ing end. The consideration of a mixed boundary conditionallows us to discuss the statistical weight of less favorableshapes. We also found that the subsection of a DNA withmultiple kinks can be described using mixed boundarycondition [37]. Several proteins studied to date are shownto induce bends and kinks on DNA molecules upon bind-ing [38–40]. During the measurement of elastic propertiesof DNA molecules, the whole chain is expected to adapt itsconformation towards the optimal energy conformation byarranging each subchain between the neighboring kinks inskew-symmetric shape. The relative orientations of bondvectors at kinks are fixed by geometrical constraint set bychemistry.

If the boundary condition with freely rotating ends ap-plies to both edges where the pulling force is exerted, theend-to-end vector is parallel to the external force (vanish-ing external torque condition). By the same reasoning, theinternal inflexion points are also on the end-to-end line.Furthermore, if the sticking proteins are evenly spaced,inflexion points will be located in the middle of each sub-chain [41]. Under the mixed boundary condition, the re-sulting shapes are not usually symmetric. This also appliesto the “equilibrium” shape where the force along the end-to-end vector vanishes. A relevant shape can be gained bya proper cut under an angle φ from a symmetric free ori-enting shape starting at one edge (see Fig. 9). Equationsvery similar to equations (8, 9) can be derived:

1

2kBT lp

(

dθ

ds

)2

+ τ cos(θ(s) + φ) = c. (27)

First, rather than using the two components of the forceµ and ν, one may express the torque acting on the in-finitesimal contour length ds with the magnitude of theforce τ and its orientation φ as τ sin(θ + φ)ds (replacingthe former f sin θds). It is also more convenient to choosethe origin for the arc length at the freely orienting edge.


tan φ = Γ (m) =2√

m√

1 − m sd`

πp

τ/fc, m´

2E`

πp

τ/fc, m´

− 2m sn`

πp

τ/fc, m´

cd`

πp

τ/fc, m´

− πp

τ/fc

. (31)

This leads to the orientation

sin((θ + φ)/2) =√m cd(u,m),

cos((θ + φ)/2) =√

1 −m nd(u,m).(28)

The angles θ(s) can be expressed in terms of equa-tions (28).

sin θ = 2

(

sin

(

θ + φ

2

)

cos

(

θ + φ

2

)

cosφ

− cos2(

θ + φ

2

)

sinφ+sinφ

2

)

cos θ = 2 cos2(

θ + φ

2

)

cosφ

+ 2 cos

(

θ + φ

2

)

sin

(

θ + φ

2

)

sinφ− cosφ

(29)

and the filament shape:

x(s) =S

π

√

fc

τ

(

− 2 sinφ√

m(1 −m) sd(u,m)

+ 2 cosφ (E(u,m)−m sn(u,m) cd(u,m)))

−s cosφ,

y(s) =S

π

√

fc

τ

(

2 cosφ√

m(1 −m) sd(u,m)

− 2 sinφ (E(u,m) −m sn(u,m) cd(u,m)))

+ s sinφ,

(30)

where m = sin2((θ0 + φ)/2), u = πsS

√

τfc

, and sn(u,m),

cn(u,m), dn(u,m) are Jacobian elliptic functions. Theconstraint y(S) = 0 links the angle φ and the parame-ter m of the Jacobian elliptic functions for the given forcemodulus τ .

See equation (31) above

The angle at the edge s = S is imposed as β:

cosβ=cosφf(m)=cosφ

(

2(1−m) nd2

(

π

√

τ

fc,m

)

− 1

+2√m√

1 −mΓ (m) nd

(

π

√

τ

fc,m

)

cd

(

π

√

τ

fc,m

))

.

(32)

Following from the two constraints equations (31, 32),the parameter m of the Jacobian Elliptic functions satis-fies

cos2 β(

1 + Γ 2(m))

= f2(m). (33)

All quantities of interest can now be calculated. For exam-ple, we get the free angle θ0 from equations (29) as sin θ0 =sinφ(2m− 1 + 2

√m√

1 −m/Γ (m)). The bending energy

is now related to τ via Ub = τ(cosφX−S cos(θ0 +φ)) andthe total energy via Ut = Ub + τ cosφX.

When the imposed angle is non-zero, the neutral shapefx = 0 is buckled, for our example (β = π/6), the freeangle is θ0 = 0.260 and the end-to-end distance 0.973S(√

2fyS2/(lpKBT ) = 1.79). The magnitude of the free an-gle is always smaller than β for the neutral shape. Undercompression the magnitude of the free angle increases andreaches that of the fixed angle. Here we recover a symmet-ric shape with an inflexion point at both edges and fy = 0.Upon further compression the inflexion point moves fromthe fixed angle edge inward. The force is directed along theline joining the internal inflexion point and the free edge.

When β = 0, there is buckling transition at a largercompressive force than for the freely rotating edges caseµc > fc. In order to find the critical compression force,

we linearize equation (5): lpd2θ(s)ds2 + µθ(s) + ν = 0. The

arclength position at s is x(s) =∫

(1 − 12θ

2(s))ds and

y(s) =∫

θ(s)ds. The Lagrangian multiplier µ is the com-pressive force in the x-direction. With the mixed boundarycondition, we require ∂θ/∂s|0 = 0 and θ(S) = β. The so-lution to the differential equation has the following form:θ(s) = A cos(s

√

lp/µ) + C. Together with the constrainty(s) = 0, we obtain the buckling force µc as the smallestvalue of force µ satisfying the condition tan(x) = x with

x = π√

µ/fc.

µc = 2.046fc = 20.19kBT lp/S2. (34)

The critical compressive force µc under mixed boundarycondition is about twice as large as fc. This result agreeswith the critical buckling force observed by Janson andDogterom [17].

In Figure 10, we represent the end-to-end distance(X = x(S)) as a function of the longitudinal force τ cosφfor two values of the imposed angle: β = 0, β = π/6.One of ends can rotate freely in order to minimize thebending energy. Two types of force-extension measure-ment can be considered: i) a compressive force is givenand ii) the total extension is controlled. With a vanishingforce fx, the filaments are at extension X0 = 0.973S forβ = π/6 and at X0 = S for β = 0. For β = 0, the filamentbuckles at µc. As the compressive force increases, the ori-entation of the freely rotating end θ0 increases with thetilted angle φ. For φ 6= 0, the end with fixed orientation βexperiences a net torque which should be balanced withthe vertical component of the external force fy. The freeend of the chain escapes from the clamping edge unlessthere is force holding the end. The filament shapes arestable as long as the extension is a decreasing function offx. For a large tilted angle φ (beyond B, G), the buckledshapes become unstable. Under the constant force, the fil-ament undergoes transition to the solutions in differentbranches. The transition is indicated as arrows in vertical


-4 -2 0 2 4fx/f

c

-0.5

0

0.5

1

X

-4 -2 0 2 4fx/f

c

-0.5

0

0.5

1

X -1 -0.5 0 0.5 1-1 -0.5 0 0.5 1

A

B

D

E

G

H

I

J

-0.5 -0.4 -0.3 -0.2 -0.1 0.1

-0.2

-0.15

-0.1

-0.05

0.05

0.1

0.15

0.2

β

β

ββ

β

β

β

-0.4 -0.3 -0.2 -0.1 0.1 0.2

-0.2

-0.15

-0.1

-0.05

0.05

0.1

0.15

0.2

(c) β=π/6 (d) β=0

ββ

H’C’

β

Α(a)

(b)

G’ J

β=π/6

β=0

I

D

B’

B

G

H’H

EF

C’C

twisting

Fig. 10. Force-extension relations of a filament under mixed boundary condition. One chain end at origin (X = 0) can rotatefreely (clamped) and the other end is imposed to start with an angle β = π/6 (top panel, a) and β = 0 (bottom panel, b). Theforce-extension curves of freely rotating ends are shown as dashed lines for comparison. Typical filament shapes at the markedpositions are shown in (c) and (d). The thin solid lines represent a path with higher energy conformations. The dot-dashed lineis obtained when twisting is allowed. The fixed angle β is indicated for each shape in panel (c) and (d).

direction (B → B′ and G → G′). Increase of fx straight-ens the filament further. However, inflexion point can nolonger exist at boundary beyond F, J.

During the force measurement under controlled exten-

sion, the force-extension path may depend on the con-straint whether the filament is forbidden to twist. If oneend is imposed to start with an angle β = π/6, the force-extension curve follows the solid line connecting (A, B) atlarge extensions. A transition to lower energy conforma-tion occurs (θ0 → π − θ0) if twisting is allowed. In Fig-ure 10(a), such a transition under a given extension isindicated by the dashed horizontal arrows. After the tran-sition, the necessary force to keep the filament at the givenposition switches the sign. The negative value of fx indi-cates that the force is now “pulling” the end. The exten-sion goes through the origin with increasing force followingthe lower curve (D → E → F). If twisting is not allowed,the path remains on the line connecting (A, B, C) untilthe hairpin-like conformation is formed. Then the pathjumps from C to C′ where the conformation switches viarotation with respect to the inflexion point.

For β = 0, the filament remains on the solid line (G)until the extension vanishes. Then lower energy config-urations are found at lower curve connecting (H, I, J).The transition to H (indicated by the arrow in Fig. 10(b))is expected at X = 0 via rotation around the inflectionpoint (without twisting). On further squeezing, the pathremains on the thick solid line (H, I, J). The thin solid linerepresents conformations at intermediate extensions suchas displayed in panel (d) as H′. These are energetically less

favorable and their contribution to the equilibrium elasticresponse is expected to be negligible.

3.6 Asymmetric fixed-angle boundary conditions

It is worthwhile to briefly discuss the case when the angleis imposed at both edges at θ1 and θ2. The main differencewith the mixed boundary is that the shift in the argumentof Jacobian Elliptic function c is not a quarter period. It isto be determined from the known imposed angle θ1 and θ2,

cos

(

θ + φ

2

)

= dn(u+ c,m), (35)

sin

(

θ + φ

2

)

=√m sn(u+ c,m), (36)

with u = sπ/S√

τ/fc. After integration, we get

x =

(

2S

π

√

fc

τ

(

E(u+ c,m) − E(c,m))

− s

)

× cosφ− 2√m sinφ

S

π

√

fc

τ(cn(u+ c,m) − cn(c,m)) ,

y = −2√m cosφ

S

π

√

fc

τ(cn(u+ c,m) − cn(c,m))

+ sinφ

(

s− 2S

π

√

fc

τ

(

E(u+ c,m) − E(c,m))

)

.

(37)


The expression for a tilt angle φ can be represented interms of τ , m and c from the constraint y(S) = 0.

tanφ = Γa(u+ c,m) =

2√m√

fc

τSπ (cn(u+ c,m) − cn(c,m))

s− 2√

fc

τSπ

(

E(u+ c,m) − E(u,m))

. (38)

By inserting equations (36) to equation (29), similarly toequation (32), we construct fa(u+c,m) = 2 dn2(u+c,m)+2√mdn(u + c,m) sn(u + c,m)Γa(u + c,m) − 1. At the

boundaries s = 0 and S,

cos2 θ1(

1 + Γ 2(c,m))

=f2a (c,m), (39)

cos2 θ2(1+Γ 2(π√

τ/fc+c,m))=f2a (π√

τ/fc+c,m).(40)

The unknown variables m, c and φ can be obtained fromequations (38–40).

4 Conclusion

Unlike usual flexible polymers that coil under thermalnoise, rigid filaments do by definition share many proper-ties of macroscopic thin rods in a first approximation. Onthe other hand, only in extreme cases like microtubules ormultiwall carbon nanotubes are shape fluctuations com-pletely negligible for standard filament length. We triedto adapt the description of optimal rod shapes undercompression to thin long rigid filaments. These are de-scribed by a persistence length lp, over which the orienta-tion correlations decay, markedly larger than the filamentlength S.

Several boundary conditions are considered. Two seemmost relevant: the free-orienting-end case and the mixedboundary case where the orientation is fixed at one end.The former applies to filaments manipulated by opticalor magnetic tweezers and the latter to many experimentswhere one end orientation is tuned. Deposition of filamentsalso sometimes falls in this category when the first ad-sorbed section (end) cannot change the orientation any-more. Fluctuations also control cyclization or zipping re-actions of filaments, or some aspects of their depositionon striped surfaces as discussed above.

As also shown earlier [13] a free filament stores a frac-tion of its length, proportional to S/lp in fluctuations.Under a compressive force, as considered here, the storedlength increases and saturates at the buckling transition.In the strongly buckled states, fluctuations around the(one of the two) more favorable shape(s) are again weakerand essentially ∝ S/lp. We estimated fluctuation correc-tions to the force-extension curves. In two dimensions asconsidered here, the end-to-end distance under a givenforce is decreased by fluctuations in the unbuckled stateand increased in the buckled state. The force-extensioncurve rounded by fluctuations hence crosses the Eulercurve in the transition region. This crossing is at leastqualitatively described by a calculation to quartic order.

As a consequence the force needed to impose average clos-ing (〈x〉 = 0) is larger than anticipated from Euler elas-tica. As shown explicitly in [24] for fixed boundary angles(also by simulation), if weak uncoupled in-plane and out-of-plane fluctuations are allowed (3-d), the end-to-end dis-tance is decreased by fluctuations also in the buckled state.The, somewhat surprising, increase of the end-to-end dis-tance in buckled state hence seems to be a peculiarity of2-d filaments. The energy barrier against closing with freeor prescribed angle is increasing with stiffness as ∝ lp/S

2

in thermal units, fluctuation corrections are estimated tobe of order unity and become important for moderatelyrigid filaments. Similar conclusions hold for zipping. In thecase of fixed boundary angles we showed that for 0 < θ0 <π/2 a looped conformation may form under compressionby twisting this is favoured by loose boundary conditionsfor the torsion angle and/or weak torsion modulus.

This work is supported by the Korea Science and Engi-neering Foundation (KOSEF) grant (R01-2007-000-10854-0)(N.-K.L.). S.-C.H. acknowledges the financial support from theKorea Research Foundation (KRF) funded by the Korean gov-ernment through grant (MOEHRD, KRF-2006-C00132) andKorea Science and Engineering Foundation (KOSEF) grant(R01-2007-000-11674-0).

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Compressing a rigid filament: Buckling and cyclization