Statics and dynamics of the wormlike bundle model.

Abstract

Bundles of filamentous polymers are primary structural components of a broad range of cytoskeletal structures, and their mechanical properties play key roles in cellular functions ranging from locomotion to mechanotransduction and fertilization. We give a detailed derivation of a wormlike bundle model as a generic description for the statics and dynamics of polymer bundles consisting of semiflexible polymers interconnected by crosslinking agents. The elastic degrees of freedom include bending as well as twist deformations of the filaments and shear deformation of the crosslinks. We show that a competition between the elastic properties of the filaments and those of the crosslinks leads to renormalized effective bend and twist rigidities that become mode-number dependent. The strength and character of this dependence is found to vary with bundle architecture, such as the arrangement of filaments in the cross section and pretwist. We discuss two paradigmatic cases of bundle architecture, a uniform arrangement of filaments as found in F -actin bundles and a shell-like architecture as characteristic for microtubules. Each architecture is found to have its own universal ratio of maximal to minimal bending rigidity, independent of the specific type of crosslink-induced filament coupling; our predictions are in reasonable agreement with available experimental data for microtubules. Moreover, we analyze the predictions of the wormlike bundle model for experimental observables such as the tangent-tangent correlation function and dynamic response and correlation functions. Finally, we analyze the effect of pretwist (helicity) on the mechanical properties of bundles. We predict that microtubules with different number of protofilaments should have distinct variations in their effective bending rigidity.

Full text

Page 1

arXiv:0909.5059v1 [cond-mat.soft] 28 Sep 2009
Statics and Dynamics of the Wormlike Bundle Model
Claus Heussinger,1,2Felix Sch¨ uller,2and Erwin Frey2
1Laboratoire de Physique de la Mati` ere Condens´ ee et Nanostructures Universit´ e Lyon 1,
CNRS, UMR 5586 Domaine Scientifique de la Doua F-69622 Villeurbanne Cedex, France
2Arnold Sommerfeld Center for Theoretical Physics and CeNS,
Department of Physics, Ludwig-Maximilians-Universit¨ at M¨ unchen,
Theresienstrasse 37, D-80333 M¨ unchen, Germany
(Dated: September 28, 2009)
Bundles of filamentous polymers are primary structural components of a broad range of cytoskele-
tal structures, and their mechanical properties play key roles in cellular functions ranging from
locomotion to mechanotransduction and fertilization. We give a detailed derivation of a wormlike
bundle model as a generic description for the statics and dynamics of polymer bundles consisting of
semiflexible polymers interconnected by crosslinking agents. The elastic degrees of freedom include
bending as well as twist deformations of the filaments and shear deformation of the crosslinks. We
show that a competition between the elastic properties of the filaments and those of the crosslinks
leads to renormalized effective bend and twist rigidities that become mode-number dependent. The
strength and character of this dependence is found to vary with bundle architecture, such as the
arrangement of filaments in the cross section and pretwist. We discuss two paradigmatic cases of
bundle architecture, a uniform arrangement of filaments as found in F-actin bundles and a shell-like
architecture as characteristic for microtubules. Each architecture is found to have its own universal
ratio of maximal to minimal bending rigidity, independent of the specific type of crosslink induced
filament coupling; our predictions are in reasonable agreement with available experimental data for
microtubules. Moreover, we analyze the predictions of the wormlike bundle model for experimental
observables such as the tangent-tangent correlation function and dynamic response and correlation
functions. Finally, we analyze the effect of pretwist (helicity) on the mechanical properties of bun-
dles. We predict that microtubules with different number of protofilaments should have distinct
variations in their effective bending rigidity.
PACS numbers: 87.16.Ka,87.15.La,83.10.-y
I. INTRODUCTION
Bundles of filamentous polymers like F-actin form pri-
mary structural components of a broad range of cy-
toskeletal structures including stereocilia, filopodia, mi-
crovilli, cytoskeletal stress fibers, or the sperm acrosome.
Actin-binding proteins allow the cell to tailor the dimen-
sions and mechanical properties of the bundles to suit
specific biological functions. In particular, the mechani-
cal properties of these bundles play key roles in cellular
functions ranging from locomotion [1, 2, 3] to mechan-
otransduction [4] and fertilization [5]. In view of this
ubiquity, a detailed understanding of bundle mechan-
ics is fundamental to gaining a mechanistic understand-
ing of cellular function [6]. Quantifying the governing
mechanical principles of these fundamental cytoskeletal
constituents could also prove valuable in the design of
biomimetic nanomaterials.
In-vitro experiments recently investigated the role of
actin-binding proteins like fascin, α-actinin and I-plastin
in mediating bundle mechanical properties [7]. Already
an inspection of bundle conformations from fluorescence
microscope images makes it evident that the properties of
the various crosslinking proteins must be quite distinct.
While bundles formed by fascin show a compact form and
remain straight over several microns, bundles formed by
α-actinin or filamin are wiggly and very lose [8, 9, 10].
The mechanical properties of actin bundles formed by
different crosslinking proteins was quantified by a fluctu-
ation analysis [7], which measures the magnitude of their
thermal fluctuations.It was found that the apparent
bundle bending stiffness can be varied over a substantial
range by changing the type and relative concentration of
the crosslinker.
FIG. 1: Wormlike bundle model. We consider bundles that
consist of regular arrangements of filament. These are as-
sumed to be locked in place by crosslinking proteins. When
the bundle bends and twists in space, the filaments start to
slide along each other. This effect leads to shear deformation
in the crosslinks.

Page 2

2
These intriguing mechanical properties can be under-
stood in terms of the wormlike bundle model (WLB),
which describes bundles as an assembly of semiflexible
filaments interconnected by crosslinking proteins [11, 12];
for an illustration of the bundle architecture see Fig. 1.
Unlike the standard wormlike chain model (WLC) [13,
14], the wormlike bundle model (WLB) exhibits a state-
dependent bending stiffness [11] that derives from a
generic competition between the bending and twist stiff-
ness of individual filaments and their relative motion me-
diated by the stiffness of the crosslinkers. An important
aspect of the WLB model is that crosslinks may be very
efficient in constraining the lateral excursions of filaments
within the bundle but much less so in inhibiting their ax-
ial motion. This is possible as the relative axial sliding
of two crosslinked filaments probes not only the elastic
properties of the crosslinking protein but also those of
the binding domain at which the protein is attached to
the filament. The latter may be of quite different rigidity
than the protein itself. Unfortunately there are no single-
molecule experiments yet which would quantify the me-
chanical and binding properties of actin crosslinking pro-
teins attached to a pair of F-actin filaments [62]. In the
WLB model the mechanical properties of the crosslinks
are described by a single shear-stiffness k×for the relative
sliding of the constituent filaments.
An important finding within the wormlike bundle
model is that the mechanical properties of bundles can
be classified into three distinct bending regimes that are
mediated by both crosslink type and equally importantly
by bundle dimensions, namely diameter and length [11].
Taking into account the mechanical properties of fila-
ments and crosslinker at a microscopic level is a virtue of
the model making it quite verstile. It reduces to a well de-
fined continuum limit but is equally applicable to bundles
with as few as two filaments. This microscopic perspec-
tive may prove a valuable starting point to address more
complex questions related to bundle mechanical proper-
ties. This may include problems such as disorder, lattice
defects [15], or filament fracture [16].
While previously we have provided a formulation only
for plane-bending in two dimensions [12], here we give
a full derivation of the WLB Hamiltonian in three spa-
tial dimensions. This includes bending as well as twist
deformations. We explore the predictions of the WLB
model for experimental observables such as the tangent-
tangent correlation function and dynamic response func-
tions. Moreover, we discuss the effects of different bun-
dle geometries on their mechanical properties. In this
respect, we will view microtubules as a bundle of (proto-
)filaments arranged on the surface of a cylinder; compare
Fig. 2. This shell-like bundle architecture is contrasted
with a uniform distribution of filaments as found for F-
action bundles [17, 18]. Finally, we discuss how helicity
influences bundle mechanics.
FIG. 2: llustration of the bundle geometries considered. The
position of the kth filament in the cross-sectional plane ((y,z)-
plane) is given by Rk. F-actin bundle architecture (left): fil-
aments are arranged on a square lattice. Microtubule archi-
tecture (right): filaments are arranged on the surface of a
cylinder.
II. MODEL DEFINITION
We consider bundles of length L that consist of N par-
allel filaments. While the filaments may form a disor-
dered lattice structure, we will focus our attention here
to the cases of regular arrangements. In particular we
will treat in detail the square-lattice and the cylindrical
tube (see Fig. 2).
Each filament is modeled mechanically as an exten-
sible worm-like polymer with stretching stiffness ks,
bending rigidity κb and twist rigidity κt.
are irreversibly crosslinked to their nearest-neighbors by
crosslinks with mean axial spacing δ.
are modeled to be compliant in shear along the bundle
axis with finite shear stiffness k×, and to be inextensible
transverse to the bundle axis, thus constraining the in-
terfilament distance, b, to be constant (see Fig. 2). This
assumption, which neglects crosslink stretching deforma-
tions, is based on the recognition that the shearing mode
involves deformation of the crosslink and its binding do-
main to the filament. The resulting stiffness may indeed
be much lower than that of a crosslink in isolation.
Filament stretching is characterized by the axial dis-
placement uk(s) of filament k at axial position s along
the backbone. To describe bundle bending and twisting
we define {d1,d2,d3} to be a material frame fixed to
the bundle central line at each arclength position s. The
vector d3 ≡ t is the tangent to the space curve traced
out by the central line, while the two vectors d1and d2
lie within the cross-section of the bundle. The position
of each filament in the cross-section is parametrized in
terms of a vector Rk(s) = Akd1(s) + Bkd2(s), where
Filaments
The crosslinks

Page 3

3
Ak and Bk are the material-frame coordinates of fila-
ment k; they are constants independent of arclength s
and deformation of the central line (see Fig. 2). As one
moves along the bundle backbone the material frame ro-
tates according to Frenet-Seret equations, ∂sd = Ω × d.
The rate of rotation is given by the generalized curva-
tures Ω = (Ω1(s),Ω2(s),Ω3(s)), which, in addition to
the axial displacement uk, represent the basic kinematic
degrees of freedom of the bundle.
A. The WLB Hamiltonian
Neglecting all nonlinear effects we are now going to
develop a simple expression for the bundle energy which
is harmonic in its degrees of freedom, axial displacement
ukand generalized curvatures Ωα.
This WLB Hamiltonian consists of three contributions,
HWLB = H0+ Hstretch+ Hshear. The first term corre-
sponds to the standard wormlike-chain Hamiltonian
H0=N
2
?L
0
ds?κb
?Ω2
1+ Ω2
2
?+ κtΩ2
3
?. (1)
Writing this we assume that each filament follows effec-
tively the same space-curve as the center-line. While,
in general, one should account for the curvatures, Ωk,
of each individual filament separately, this would only
lead to correction factors that can be neglected for our
purposes. Consider, for example, planar bending of the
central line, Ω1 = 1/ρ, where ρ is the radius of curva-
ture. The filaments that lie at a distance R away from
the central line naturally have a different radius of cur-
vature, ρ ± R, and thus a different bending energy. The
magnitude of the correction term relative to Ω2
since it scales as (R/ρ)2, i.e. we assume the typical cur-
vatures to be smaller than the bundle radius (for more
details see Appendix B).
As to the twist degree of freedom, in Eq. (1) we do
not allow for the possibility of relative twisting of the
individual filaments (see Section V and Appendix B for
a discussion of this effect). We assume that twist is only
due to the “bundle-twist” Ω3 of the central line. This
assumption is reasonable in tightly bound bundles, where
the filaments are connected by many crosslinks. In this
state the filaments and their relative orientation can be
assumed to be locked-in by the crosslinker.
The second term in the Hamiltonian, Hstretch =
?
crosslinks at arclength positions si and si+ δ, respec-
tively.
1is small
kHk
stretch, accounts for filament stretching. It depends
on the difference in axial displacement, uk, between two
Hk
stretch =
ks
2
?
?L
i
[uk(si+ δ) − uk(si)]2
→
ksδ
2
0
ds
?∂uk
∂s
?2
,(2)
where we have performed the continuum limit?
ks(δ) is the single filament stretching stiffness on the scale
of the crosslink spacing δ.
The particular form for ks depends on the system
under consideration. For high crosslink concentrations
(small δ), the segment behaves as a homogeneous elas-
tic beam, characterized by a Youngs modulus E and
kbeam
s
∼ Eb2/δ. The combination ksδ that enters the
Hamiltonian is independent of δ, as it should: the me-
chanical stretching stiffness of a beam cannot depend on
the properties of the crosslinks.
For small concentrations of crosslinks (large δ) entropic
effects become relevant and the stretching stiffness is that
of a thermally fluctuating wormlike chain with persis-
tence length lp. In this case one has kentr
which implies that the combination ksδ ∼ δ−3does de-
pend on the crosslink spacing δ. This is related to the
fact that the formation of a crosslink suppresses ther-
mal undulations (reduces entropy) and thus increases the
entropic stretching stiffness. Equating both stretching
stiffnesses, kbeam
s
∼ kentr
concentration, δ3
enthalpic to entropic elasticity takes place.
In the case of microtubules, which will be treated in
Section IV, the crosslink spacing δ is given by the tubulin-
size and the stretching stiffness is modeled as for an elas-
tic beam.
The third energy contribution, Hshear =?
ing filaments. The relative axial motion of a filament
pair (l,k) at a given point of the backbone is described
by the crosslink shear displacement, which is the sum of
a geometric contribution, ∆lk, and the relative stretching
of neighboring filaments, ∆ulk= ul− uk. The geomet-
ric part results from the arclength mismatch between the
two filaments, induced by a bending and twisting of the
bundle central line. As in Eq. (2) we first write the shear
energy as a sum over all crosslink positions siand then
perform the continuum limit, to get
i→
?ds/δ to arrive at the second line. The spring constant
s
∼ κblp/δ4,
s
one finds the critical crosslink
c∼ b2lp, at which the cross-over from
lkHlk
shear,
results from the crosslink-induced coupling of neighbor-
Hlk
shear =
k×
2
?
?L
i
[∆lk(si) + ∆ulk(si)]2
→
k×
2δ
0
ds(∆lk+ ∆ulk)2,(3)
where k×is the shear stiffness of the individual crosslink.
For any bundle deformation, the associated value of
∆lkcan be compensated for by stretching the filaments,
making the shear energy vanish when ∆ulk= −∆lk. At
the same time, however, this would increase the stretch-
ing energy, which may be unfavorable if the stretching
stiffness ksis rather large. For deformations on the scale
of the bundle length (u′∼ u/L) the ratio of both en-
ergies gives the important parameter α = k×L2/ksδ2,
which quantifies the relative strength of both deforma-
tion modes [11].
As a final ingredient to the model we need to calculate

Page 4

4
the dependence of ∆lk on the bundle curvatures, Ωα.
Without going into the details of an explicit derivation,
we here just give the resulting expression. For more de-
tails we refer the reader to Appendix C. The special case
relevant for the microtubule geometry is also dealt with
in some detail in [19]. To linear order in Ωαwe find
∆lk = blkcosαlk
?
ylkΩ3−
?
?s
?s
0
dtΩ2(t)
?
(4)
−blksinαlk
zlkΩ3−
0
dtΩ1(t)
?
,
where we defined blk= |blk| as the distance between the
filament pair and αlk as the angle of blk with respect
to the z-axis. Furthermore, (ylk,zlk) are defined as the
cross-sectional coordinates of the midpoint between the
filament pair.
In contrast to H0, which is an expansion in the general-
ized curvatures Ωα, the shear energy Hshearis a function
of the integrated curvatures, since ∆ ∼ b?
one has to assume that the shear displacement is suffi-
ciently small, ∆ ≪ a, where a is some microscopic length-
scale related to the size of the crosslink. In terms of the
bundle curvatures this implies LΩ ≪ a/R ∼ 1, which is
much more restrictive than the range RΩ ≪ 1, overwhich
H0can be approximated by a harmonic form. As a con-
sequence the bundle is only allowed to make small excur-
sions from its initial state, an assumption which is usu-
ally well satisfied in bundles of stiff polymers like actin or
microtubules, but certainly breaks down under extreme
loading conditions (e.g. to describe post-buckling) or in
more flexible objects like DNA.
With this “weakly-bending” assumption we reformu-
late the generalized curvatures in terms of the lab-frame
Euler angles [20, 21]
sΩ ∼ bLΩ. For
terms beyond the harmonic contribution to be negligible,
Ω1 =
∂φ
∂ssinψ sinθ +∂θ
∂φ
∂scosψ sinθ −∂θ
∂ψ
∂s+∂φ
∂scosψ
Ω2 =
∂ssinψ
Ω3 =
∂scosθ .
As reference state we take φ0 = π/2, θ0 = π/2 and
ψ0 = sω0, which corresponds to a straight, but pre-
twisted bundle that points along the x-axis. For small
excursions around this reference state we can linearize
the equations such that
Ω1 =
∂φ
∂ssinψ0+∂θ
∂φ
∂scosψ0−∂θ
∂ψ
∂s,
∂scosψ0
(5)
Ω2 =
∂ssinψ0
Ω3 =
and the angles are now measured relative to the reference
state. As expected the pretwist ψ0 leads to a coupling
of the angles θ and φ. In the following we are primarily
concerned with the case of vanishing pretwist. Then, the
coupling terms vanish and we can simply set
(Ω1,Ω2,Ω3) = (θ′,φ′,ψ′).
We will come back to the case of pretwist in Section V.
B. Examples for the arclength mismatch ∆
For the purpose of illustration we provide some ex-
amples of how the shear displacement ∆ depends on the
geometry of the bundle cross-section and the deformation
of its central line.
If the bundle consists of only two filaments [22] (ge-
ometry of a ribbon) we have y = z = α = π/2 as the
central line and the mid-line between the two filaments
are identical. In this case ∆ simplifies to
∆ = bθ(s), (6)
which is illustrated in Fig. 3. Note, that twist (Ω3) does
not contribute, as both filaments twist around the central
line symmetrically.
u1
u2
b
b
θ
θ
FIG. 3: Illustration of crosslink shear deformation for the case
of a two-filament bundle (filaments in red). Bundle deflection
through the angle θ leads to the arclength mismatch, ∆ = bθ.
The filaments have to stretch the relative amount u1−u2 = bθ,
in order to keep the crosslink (dashed line) undeformed with
zero shear energy.
A particular case of this two-filament bundle has been
considered in a set of articles [23, 24, 25], where the
crosslinks are assumed to be rigid with respect to shear,
i.e. k×→ ∞. To satisfy a vanishing arclength mismatch
∆ one thus requires Ω1≡ −d′
the unit vector d2, which points from one filament to
the other, must not rotate in the direction of the tan-
gent t. In other words, d2must equal the the bi-normal.
Thus the ribbon orientation is completely specified by
the space curve traced by the central line.
A second example is the axoneme in eukaryotic flag-
ellae [19]. There, filaments (microtubule doublets) are
arranged on the surface of a cylinder, just as the protofil-
aments in a single microtubule. Switching to polar coor-
dinates in the cross-sectional plane (see Fig. 2), (y,z) →
(R,ϕ), α = ϕ + π/2, we find
2· t = 0. This means that
∆ = −bRψ′+ bφsinϕ + bθcosϕ.(7)

Page 5

5
RΩ
RΩ
b
t
b
3
3
FIG. 4:
twisted microtubule. Indicated (in red) is a filament pair that
winds around the microtubule cylinder (radius R) taking an
angle RΩ3 with the cylinder axis t. The resulting arclength
mismatch is given by ∆ = bRΩ3.
Illustration of crosslink shear deformation for a
A similar expression, disregarding the possibility of twist-
ing, has been given by Mohrbach et al. [26]. The struc-
ture of the second and third term is the same as in
Eq. (6), additionally taking into account the modified
orientation in the cross-sectional plane, as described by
the angle ϕ. The origin of the first term is illustrated in
Fig. 4 and elaborated on in Appendix C.
Consider now the case of planar bending. This will
make the connection to continuum elasticity particularly
clear. Under planar deformation the bundle is described
by the one variable θ′= Ω such that the shear defor-
mation is simply ∆ = bθ(s) ∼ b∂xuy. Here, uy is the
displacement of the bundle transverse to the bundle axis
(in the y-direction). This latter form makes clear that
the shear deformation represents one part of the strain
tensor component ǫxy =
part, ∂yux∼ b∆ulkis the continuum version of relative
filament stretching (see Eq. (3)).
The elastic symmetries relevant to the WLB model
depend on the arrangement of the filaments in the cross-
section. If there is rotational symmetry with respect
to the bundle axis, one speaks of transversly isotropic
elastic bodies [27]. While, in general, this has five first
order elastic constants, our model has only two, aug-
mented by the (second order) bending/twisting elasticity
of the individual filaments which is not accounted for in
continuum elasticity. The simplification arises from as-
suming transverse inextensibility as well neglecting cross-
sectional shape changes. The latter is, for example, im-
portant in the failure of hollow tubes under bending. One
relevant effect is the Brazier effect [28], which describes
the increasing ovalization of the cross-section under the
1
2(∂xuy+ ∂yux). The second
action of a bending moment. In the present formula-
tion of the model these nonlinearities are not accounted
for. For a discussion of potential modifications to include
cross-sectional deformations, we refer the reader to the
outlook section at the end of this article.
C.Definition of effective bending and twist
rigidities
It should be clear from the way the Hamiltonian was
derived that the model is applicable to bundles with ar-
bitrary (ordered/disordered) arrangements of filaments
in the cross-section. In the remainder of this article we
will focus our attention to bundles with highly symmetric
cross-sections, where the filaments either form a rectan-
gular array or a hollow tube.
In view of recent experiments probing the mechanical
or statistical properties of individual bundles in vitro [7],
we head at a description of the bundle in terms of effec-
tive bending and twist rigidities. These are defined with
respect to the standard worm-like chain model. To ar-
rive at the proper expressions we have to integrate out
the internal stretching variable u, which in general is not
observable in experiment.
To show how this works, we symbolically write the
partition function as Z =
nifies the set of Euler angles φ(s),θ(s),ψ(s).
constrained partition function then reads Z(φ)
?D{u}exp(−βH({φ,u})) ≡ exp(−βW(φ)).
formed. As the Hamiltonian is harmonic we are left with
only Gaussian integrals, which are evaluated in Fourier
space. The resulting potential of mean force W(φ) can
be written in the form of a wormlike chain Hamiltonian
?DφZ(φ) where φ sig-
The
=
The integration over the u-variables can easily be per-
W(φn) =L
4
?
n
q2
n
?κB(n)(φ2
n+ θ2
n) + κT(n)ψ2
n
?
(8)
with effective bending and twist rigidities κB(n) and
κT(n), respectively. We note that in the symmetric sit-
uations considered here there is no coupling between the
different deformation modes bending and twisting (see
Appendix A). In contrast to the usual WLC, the effec-
tive bend and twist rigidities are in general dependent on
the mode-number n and thus on the wavelength of the
deformation. This effect and the discussion of its conse-
quences is the central topic of the remaining sections.
III.F-ACTIN BUNDLE ARCHITECTURE
In the following sections we will focus our attention
to bundles with N = (2M)2filaments that form a rect-
angular array (see Fig. 2). The angle α that specifies
the orientation of the filament-pair in the cross-section is
then α = 0,π/2 as filaments are either arranged along the
y- or the z-axis. As mentioned above the different defor-
mation modes decouple in harmonic order. We can thus

Page 6

6
investigate bending independently from twisting. Also
the two space-directions decouple and we can reduce the
model to an effective two-dimensional description [12].
A. Effective bending rigidity
The shear Hamiltonian reduces to
Hshear=Mk×
δ
?L
0
ds
M−1
?
k=−M+1
(uk+1− uk+ bθ)2, (9)
where we used Eq. (4) with φ = ψ = 0. By following
the recipe outlined above we eliminate the axial strain
variable uk. By approximating ukby a linearly increas-
ing function of k (see discussion below) we arrive at the
following result for the effective bending rigidity [63] as
defined in Eq. (8)
κB(n) = Nκb
?
1 +
?12ˆ κb
N − 1+ (qnλ)2
?−1?
. (10)
Here, we have defined a characteristic wavelength
λ =
?
2M
(2M − 1)·
?
κbδ
k×b2, (11)
and a dimensionless bending rigidity ˆ κb = κb/(ksδb2).
In terms of the quantities λ and ˆ κb the previously de-
fined α = k×L2/ksδ2can be rewritten as α ∼ ˆ κb(L/λ)2.
If the filaments behave as homogeneous elastic beams,
ˆ κbis just a number independent of bundle geometry or
crosslink spacing. For any numeric computation we will,
for specifity, assume that ˆ κb= 1/12, which corrresponds
to beams with square cross-sections [29].
The characteristic feature of Eq. (10), is the wave-
length dependence (see Fig. 5). For wavelengths q−1
the interval 1/√N ≪ qnλ ≪ 1 the bending stiffness de-
creases as κB(n) ∼ k×q−2
this the intermediate or shear dominated regime as the
bending rigidity is proportional to the shear stiffness of
the crosslinks. It is in this parameter regime that the
bundle behaves qualitatively different than either a ho-
mogeneous beam (obtained in the ”fully coupled” limit of
qnλ ≪ 1/√N) or an assembly of ”decoupled” filaments
(qnλ ≫ 1).
Another important feature of Eq. (10), which is in-
dependent of the specific q-dependence, is the ratio of
maximal to minimal bending rigidity, r = κmax/κmin=
1 + (N − 1)/12ˆ κb. This only depends on the number of
filaments and the dimensionless bending rigidity ˆ κb.
While integrating out the stretching variables uk can
be performed exactly, Eq. (10) is based on the addi-
tional assumption that axial strains are linearly increas-
ing through the cross-section, uk= ∆u · (k + 1/2). The
exact profile for ukis calculated in the appendix and dis-
played in Fig. (6); compared to the linear profile it shows
an enhancement of strain towards the bundle periphery.
n
in
n. In Ref. [11] we have termed
1
0.01
10
0.1 1 10
κB/Nκb
qλ
FIG. 5: Effective bending rigidity, Eq. (10), as a function
of mode-number, qλ, and for the set of bundle sizes N =
4,9,16,25 (from bottom to top). In the fully-coupled and the
decoupled regimes (corresponding to small and large q) the
bending rigidity is constant. At intermediate values of q the
bending rigidity scales as κB(n) ∼ k×q−2
regime).
n
(shear-dominated
-10
-5
0
5
10
-9-6-3 0
k
3 6 9
uk
α=10
α=105
FIG. 6: Dependence of axial strain uk on k, the distance
from the bundle central line. Decreasing the dimensionless
shear-stiffness α = k×L2/ksδ2the strain is reduced but not
in a linear fashion. The outer layers of the bundle remain
stretched stronger than the inner ones.
However, the ensuing value for the bending stiffness is
largely insensitive to the linear approximation [64]. We
speculate that the nonlinearities in the axial strain may
eventually be important for nonlinear material proper-
ties, as for example strain induced rupture.
creased strain in the outermost filaments brings them
closer to their threshold for rupture and thus makes them
more susceptible to this mode of failure.
In order to make contact with continuum models for
beam bending we perform a continuum limit, by letting
N → ∞ but keeping the bundle aspect ratio D/L ∼
bM/L constant. Then, bundle length L has to grow with
M as L(M) ∼ M. In particular, this implies that fewer
and fewer modes n belong to the decoupled regime (where
q−1
n
≪ λ). Eventually, this regime, where filaments bend
independently (κB(n) ≈ Nκb) is no longer accessible.
The in-

Page 7

7
In effect this means that the bending stiffness κbof the
individual filaments can be neglected, just as in ”nor-
mal” continuum elasticity, where higher order gradients
(O(θ′)) are not accounted for from the start.
In this continuum limit the result from the lineariza-
tion assumption, Eq. (10), reduces to the Timoshenko
model for beam bending [30], which was recently used
to interpret bending stiffness measurements on micro-
tubules [31, 32] and carbon nanotube bundles [33],
κTIMO
B
(n) =
N2κb
1 + (qnD)2E/12G.
(12)
To allow for comparison with continuum elasticity we
have used the expressions ksδ = Eb2and κb = Eb4/12
applicable for homogeneous beams of square cross-section
and defined the shear-modulus G = k×/δ.
On the other hand, one can equally derive a continuum
limit from the exact expression for κB(n) (as presented
in Appendix A1). This gives
κCONT
B
(n) =
N2κb
(qnD)2E/12G
?
1 −tanh(qnD?E/4G)
qnD?E/4G
?
(13)
.
Both expressions, Eq. (12) and Eq. (13) are compared
in Fig. (7). One infers that the approximation (Tim-
oshenko theory) overestimates the exact solution by no
more than 6%. The difference can partly be compensated
for by introducing a shear-correction factor (≈ 1.2) in the
denominator of Eq. (12).
0 0
0.2 0.2
0.4 0.4
0.6 0.6
0.8 0.8
1 1
0 0 2 2 4 4 6 6 8 8 10 10
κB/Ν2κb
xx
κB/Ν2κb
3%3%
6%6%
0 2 4 6 8 10
xx
error
0 2 4 6 8 10
error
FIG. 7: Continuum limit of effective bending rigidities as a
function of x = (qnD)2E/12G. Comparison of exact solution,
Eq. (13), with approximation, Eq. (12). Inset: Relative error
between both expressions. The approximation over-estimates
the bending stiffness by maximal ∼ 6%.
B.Tangent-tangent correlation function
In this section the implications of a mode-number de-
pendent bending rigidity is further elaborated by dis-
cussing the concept of the persistence length. The per-
sistence length of a single WLC may be defined in terms
of the competition of bending and thermal energies,
lp= κ/kBT. With this definition, bending rigidity and
persistence length are basically identical. In the frame-
work of the WLB this would lead to a mode-number de-
pendent persistence length lp(n) = κB(n)/kBT.
For the WLC the persistence length is also the length-
scale over which the tangent-tangent correlation function
decays
?t(s)t(0)? = exp[−s/lp] . (14)
This simple exponential form is no longer valid for the
WLB as can be illustrated by considering planar bending
with φ = 0. Then the tangent-tangent correlation func-
tion is easily inferred from the angular fluctuations [22]
as
?t(s)t(0)? = exp
?
−1
2
?
(θ(s) − θ(0))2??
, (15)
with
?
and A = 1 + 12ˆ κb/(N − 1).
expression into the form given by Eq. (14) implies an
arclength-dependent persistence length, lp(s). At short
distances, one recovers the decoupled regime and lp(s) =
Nlp, while at long distances, lp(s) = N2lp as found in
the fully coupled regime.
Note, that there is no immediate relation between this
lp(s) and the lp(n) defined above. The Fourier-transform
of lp(n) is, instead, given by the following expression
(θ(s) − θ(0))2?
=
1
Nlp
?A − 1
A
s +
λ
√
AA
?
1 − e−s√A/λ??
,
(16)
Forcing such a complex
l⋆
p(s) =Nκb
kBT
?
Lδ(s) +L
λ
e−√A−1s/λ
√A − 1
?
.(17)
This quantity appears in the elastic energy expressed in
real-space as
HWLB= kBT
?
θ′(s1)l⋆
p(s1− s2)θ′(s2)ds1ds2,(18)
which is a non-local function of arclength. The length-
dependence obtained here is markedly different from that
found in the correlation function, Eq. (16). While lp(s)
is constant at large distances, l⋆
tially and vanishes over the length-scale λ/√A − 1 ∼
λ?N/ˆ κb, which corresponds to the onset of the fully-
A similar nonlocal energy function is obtained when
considering the fluctuation properties of elastic mem-
branes. In-plane shear- and compression-modes lead to a
renormalized bending rigidity for the out-of-plane fluctu-
ations [34, 35]. In contrast to bundles, however, there the
kernel is long-ranged lp(s) ∼ 1/s, which asymptotically
leads to a flat membrane phase.
Concluding this section, we find that it is impossible
to speak of a single persistence length without specifying
p(s) decays exponen-
coupled regime.

Page 8

8
the precise experimental conditions as well as the observ-
able under consideration (here tangent-tangent correla-
tion function). The WLB model presents a framework
within which a length-dependent bending stiffness can be
understood. However, we would like to emphasize that
the fundamental quantity is the qn-dependent κB(n) as
presented in Eq. (10) or below in Eq. (26). It depends on
wavenumber in a universal way independent of the type
of measurement. The dependence on bundle length, in
contrast, arises through a specific transformation to real
space and may produce different results depending on the
observable under consideration.
C. Frequency-dependent correlation and response
functions
In our previous publications [11, 12] we have calculated
several thermodynamic observables and showed that the
mode-dependent bending stiffness of a WLB may lead
to drastic modifications of their scaling behavior. Here,
we widen the scope of this analyis by discussing dynamic
observables. In analogy to the usual overdamped dynam-
ics of a WLC we can discuss the dynamics of a WLB by
substituting the mode-dependent bending stiffness in the
standard Langevin equation of motion for the transverse
bending modes r⊥(qn,t)
ζ∂r⊥
∂t
= −κB(n)q4
nr⊥+ ξ(qn,t). (19)
With this one obtains for the reduced correlation func-
tion
C(t) := L−1
?L
?
0
?
(r⊥(s,t) − r⊥(s,0))2?
1 − e−t/τn
κB(n)q4
n
ds.
=
kBT
L
n
, (20)
with the relaxation times τn = ζ/κB(n)q4
(constant κB ≡ κ) one finds a scaling regime at times
t < τ1∼ ζL4/κ, where C(t) ∼ t3/4[36, 37, 38].
For WLBs, on the other hand, one has to use the q-
dependent effective bending stiffness, Eq. (10), instead.
As the term (qλ)−2multiplies the q4contribution in
Eq. (20) one finds the correlations to grow in time as
CWLB(t) ∼ t1/2. Of course, this result is only valid as
long as the qλ-term dominates the effective bending stiff-
ness, i.e. as long as the bundle is in the intermediate
regime. In fact, there will be a complex cross-over sce-
n. For WLCs
nario
C(t) ∼























(Nlp)−1
?Nκbt
?Nκbt
?ˆ κb
N,
ζ
?3/4
,t ≪ t1
(Nlp)−1λ2
ζ
?1/2
,t1≪ t ≪ t2
(Nlp)−1
N
?1/4?Nκbt
ζ
?3/4
,t2≪ t ≪ t3
(Nlp)−1L3ˆ κb
t ≫ t3
(21)
with the cross-over times t1 = ζλ4/Nκb and t2 = t1ˆ κb
governing the cross-over from the decoupled to the inter-
mediate and the fully-coupled regimes. At times larger
than t3= t1ˆ κb
N
From the correlation function one can furthermore cal-
culate the response function χ⊥ measuring the linear
response to transverse forces.
dissipation theorem and the Kramers-Kronig relations
one finds
N
L4
λ4the correlation function saturates.
Using the fluctuation-
χ⊥(ω) =
?
n
1
L
1
n− iζω.
κB(n)q4
(22)
This contrasts with the response function χ?for stretch-
ing forces
χ?(ω) =
?
n
1
lp(n)
1
κB(n)q4
n− iζω/2,(23)
which is sometimes taken as a starting point to determine
the high-frequency response of entangled solutions of stiff
polymers [39, 40, 41, 42, 43], The transverse response, on
the other hand, has been argued to relate to a “microrhe-
ological modulus” [44, 45] that is measured locally by
imbedding probe beads into the network. Unfortunately,
for a WLC both functions are hardly indistinguishable,
and only differ by the constant factor χ?/χ⊥ = L/lp.
The high-frequency behavior in both cases is χ ∼ ω−3/4.
In the case of a WLB, things are somewhat different, as
the additional factor L/lp(q) in the longitudinal response
function not only changes the prefactor but also modifies
the functional form with respect to frequency (see Fig. 8).
For similar reasons as in the discussion of the corre-
lation function one expects an intermediate regime with
ω−1/2, at least in the transverse response. This should
lead to measurable signatures in microrheological exper-
iments on bundled F-actin systems. Due to the addi-
tional q-dependence in the denominator, the longitudi-
nal response, χ?shows a smooth cross-over between the
two asymptotic regimes of fully-coupled and decoupled
bending, as explained in the figure.

Page 9

9
Ω?0.75
Ω?0.5
68 10
Ω
12 14
10?6
10?5
10?4
0.001
Χperp
Ω?0.5
Ω?0.75
68 10
Ω
1214
10?7
10?6
10?5
10?4
0.001
Χpar
FIG. 8: Comparison of transverse response function χ⊥(ω)
(top) with longitudinal response function χ?(ω) (bottom) for
a bundle of N = 20 filaments (in units of Nκb/L3and as-
suming LkBT/Nκb = 1). Frequencies are plotted in units of
ζL4/Nκb. The different curves correspond to different val-
ues of λ/L. The two asymptotic scaling regimes with ω−3/4
correspond to the decoupled (top) and fully-coupled regime
(bottom), respectively. The intermediate ω−1/2is sharper in
χ⊥ than in χ?even though the latter function shows overall
a stronger variation with frequency.
D. Effective twist rigidity
We now turn to the discussion of the twist mode. In
this case the Euler angles φ and θ in Eq. (4) are zero such
that the shear Hamiltonian reduces to
Hshear=k×
2δ
?L
0
ds
?
lk
(∆ulk+ bdlkψ′)2,(24)
where we defined the geometric factor dlk= ylkcosαlk−
zlksinαlk.
As may already be apparent from comparing Eq. (9)
with Eq. (24) the stretching deformation u couples differ-
ently to twist (k×uψ′) as to bending (k×uθ). The differ-
ence being the additional derivative occuring in Eq. (24).
Effectively this means that the resulting twist rigidity will
not depend on mode-number qn but receive a constant
correction to the single filament value κt.
Let us first assume that the filaments are inextensible.
Then the u-terms in the Hamiltonian identically vanish
and the effective twist rigidity can simply be read off from
the terms multiplying ψ′2,
κT := Nκt
?
1 +N − 1
6
?λt
b
?−2?
. (25)
Here we have defined λt:=
which is similar to λ defined above, with κbsubstituted
by κt. Unlike λ, however, it multiplies the constant
length b, the lateral distance between neighboring fila-
ments. As anticipated, there is no mode-number depen-
dence.
?2M/(2M − 1)?κtδ/k×b2,
By introducing bundle diameter D and effective shear
modulus G (as in the discussion leading to Eq. (12)) the
second term takes a form well known from continuum
theory, ∼ GD4[29]. Thus, we find an effective twist
rigidity, Eq. (25), that just describes the simple additive
superposition of two contributions, shear-induced rigid-
ity (second term) and twist rigidity of the individual fil-
aments (first term). In the continuum limit N → ∞
the latter contribution can naturally be neglected as it
only grows with N as compared to the N2increase of
the shear-induced rigidity.
Now we allow for finite axial displacements uk. This is
commonly referred to as cross-sectional warping; under
twist deformations the bundle cross-sections do not stay
plane but deform and acquire a curvature. Just as in the
case of the bending rigidity, the exact solution for the
twist rigidity only differs little from the foregoing sim-
plified analysis. Some details about the derivation are
presented in Appendix A2. The resulting axial displace-
ments uk can be found in Fig. (9) The classic solution
of Saint-Venant [27] is approached closer and closer for
increasing the shear-stiffness k×.
-150
-100
-50
0
50
100
150
-30 -20 -10 0 10 20 30
k
uk
α=10
α=105
FIG. 9: Cross-sectional warping uk (k = −M + 1...M, cor-
responding to one row of the rectangular array) for differ-
ent non-dimensional crosslink shear stiffness α = k×L2/ksδ2.
The classic solution for a beam (dashed line) is approached as
k× → ∞. In the opposite limite, k× → 0, there is no warping
and u ≡ 0, as filaments remain uncoupled in this limit.

Page 10

10
IV. MICROTUBULE ARCHITECTURE
In this section we want to turn our attention to the
case of microtubules, which we model as bundles with
filaments arranged on the surface of a cylinder. For the
time being we assume that in the groundstate the micro-
tubule is untwisted such that the N protofilaments are
oriented along the cylinder axis. This assumption is in
fact only valid for microtubules with N = 13 protofila-
ments [46, 47]. This class, nevertheless, seems to be the
most frequent in in-vitro polymerization experiments.
There is an ongoing debate in the literature about the
dependence of microtubule bending rigidity on length [31,
32, 48, 49, 50, 51, 52, 53]. Early buckling experiments
indicated a length-dependence [48], while an improved
version of the same experiments later gave a negative
result [49]. Brangwynne et al. [50] performed a mode
analysis of microtubule contours. Their data is compati-
ble with a length-dependent bending rigidity but the au-
thors vote for a cautious interpretation of the results in
view of the close proximity to the noise level. Recently,
experiments by Pampaloni et al. [32] measured the trans-
verse fluctuations of grafted microtubules to establish an
increasing persistence-length for microtubule lengths up
to L ∼ 23µm. Using a high-precision tracer technique,
Taute et al. [51] analyzed shorter microtubules and found
the persistence length to level off at lmin
lengths shorter than L ≈ 5µm. Similar values have been
obtained in Ref. [52] (lmin
p
(lmin
p
= 240µm), and explained with the help of the WLB
model. On short length-scales (in the decoupled regime)
the effective bending rigidity is constant because it re-
flects the stiffness of the individual proto-filaments. In
contrast, Kis et al. [31] have found a decreasing bending
stiffness even for microtubule sections as short as several
hundred nanometers.
While the discussion about these partially conflicting
measurements is ongoing, we would like to point out that
different techniques do not necessarily have to come to
the same conclusion, once the idea of the bending rigid-
ity as a fundamental material parameter is given up. In
the context of the WLB model the fundamental quantity
is a mode-dependent bending rigidity. As discussed in
Section IIIB, any dependence on bundle length is a “sec-
ondary” effect that will depend on the observable under
consideration.
By using the same procedure as in the case of the
rectangular bundle we find for the microtubular bend-
ing rigidity
p
≈ 580µm for
= 90µm) and in Ref. [53]
κB(n) = Nκb
?
1 +
?
8ˆ κb
sin−2(π/N)+ (qnλ)2
?−1?
, (26)
where the relevant length-scale λ is now defined as λ =
?κbδ/k×b2.
microtubule behaves as a simple beam) to scale as κ ∼
Note, that this expression results in the
bending stiffness in the fully-coupled regime (where the
N3, in contrast to bundles with a homogeneous cross-
section (as the one discussed before), where κ ∼ N2.
With Eq. (26) direct comparison with experimental
data can be made. In particular, the ratio of maximal
to minimal bending rigidity can most easily be deter-
mined, r = 1 + sin−2(π/N)/8ˆ κb= 1 + 2/sin2(π/N). For
the latter equality we assumed the protofilament to have
a circular cross-section (ˆ κb = 1/16). For microtubules
with N = 13 protofilaments this results in a universal
ratio r ≈ 35.
In this way, using the the range of values lmin
0.1...0.6mm (which we assume to represent the protofil-
ament stiffness) one can estimate the maximal persis-
tence length of long microtubules to be in the range
lmax
p
≈ 3.5...21mm. Compared with experiments [32,
54, 55] these values seem to be too large. Given the large
error-bars in any of the mentioned experiments this cal-
culation may, nevertheless, be acceptable. Furthermore,
as we will discuss below, microtubule helicity can provide
a mechanism to reduce the apparent persistence length
by reducing the effect of shear-induced coupling.
given microtubule length the persistence length is then
predicted to decrease with increasing helicity – thus im-
proving the comparison with experiment.
Finally, let’s turn to the case of pure twist deforma-
tions. Due to the symmetry of the circular cross-section
no warping is possible. We find for the microtubule twist
rigidity, similar to Eq. (25),
p
≈
For
κT := κt
?
1 +tan−2(π/N)
4
?λt
b
?−2?
, (27)
where λt:=?κtδ/k×b2.
V. PRETWISTED BUNDLES
In the previous sections we have restricted our at-
tention to non-helical bundles and assumed that in the
ground-state the filaments point along the bundle axis.
As a final application of our model, we will here discuss
the question of helicity, or pretwist, and its influence on
bundle mechanics. This aspect is important not only for
some types of microtubules but also for F-actin bundles
and is reflected in the role that helicity plays in provid-
ing an explanation for the existence of a (thermodynam-
ically) preferred bundle size [17, 56, 57].
The discussion of pretwisted bundles proceeds in two
steps. We first assume the bundle to form with all
filaments straight. Starting from this reference state
crosslink binding may add a driving force for twisting
the bundle if the straight state does not allow for opti-
mal accessibility of the crosslink binding sites. Another
effect may be the helicity of the filaments themselves that
favor a twisted bundle over an untwisted one.
Without elaborating on the precise mechanism that
leads to pretwisted bundles, we incorporate bundle
pretwist by substituting the generalized twist-curvature

Page 11

11
1
2
4
8
16
1 10
kx
100
κeff
ω0=0.25
ω0=3
FIG. 10: Effective bending stiffness (in units of Nκb) as a
function of shear stiffness k× (in units of κbδ/b2L2) of a bun-
dle of four filaments under a tip force F. The bending stiff-
ness is calculated from the determined end-deflection y(L) by
κeff = FL3/12y(L). The full curves are for finite pretwist
ω0L/π = (0.25,0.5,1,2,3), while the dashed curves represent
the limiting cases of zero and infinite pretwist, repsectively.
Ω3by Ω3−ω0. In doing so the new energetic ground-state
is at Ω1= Ω2= 0 and Ω3= ω0. To obtain the effective
bending rigidity we linearize the curvatures around this
ground-state, as performed in Eq. (5).
Inserting this result into the shear deformation,
Eq. (4), one finds terms like b?cos(ψ0)φ′ds which de-
to Fourier-space is no longer helpful as different modes
would remain coupled. We therefore resort to an alterna-
tive approach, and determine the effective bending rigid-
ity by numerical integration of the mechanical equilib-
rium equations in real space, ∂H/∂φ = 0. Specifically,
we determine the end-deflection y(L) of a bundle of 4 fil-
aments under a point force F at the distal end (s = L),
given clamped boundary conditions at the proximal end
(s = 0). The effective bending rigidity is then obtained
from the expression, κeff = FL3/12y(L), and displayed
in Fig. (10) as a function of the crosslink shear stiffness,
k×, and a series of values for the pretwist, ω0. For sim-
plicity we have assumed the filaments to be inextensible,
ks→ ∞, and thus restricted ourselves to the decoupled
and the intermediate regimes.
pend nonlinearly on arclength s. Thus a transformation
For increasing pretwist the apparent stiffness decreases
and asymptotically approaches the value without shear-
stiffness. Thus, in pretwisted, helical bundles the
crosslinks only have a limited ability to mechanically
couple the different filaments together. The higher the
twist the more the filaments act as if they were inde-
pendent [22]. The reason for this behavior is that in
pretwisted bundles the filaments exchange their place and
those that start on the top of the bundle soon are on the
bottom. Thus, crosslink sites that would stay behind and
lead to large shear displacements in untwisted bundles,
can now catch up to make the effective shear deformation
smaller.
VI.CONCLUSIONS AND OUTLOOK
We have presented a detailed study of the elastic and
dynamic properties of bundles of semiflexible filaments
(wormlike bundle model, WLB). It is found that a com-
petition between the elastic properties of the filaments
and those of the crosslinks leads to renormalized effective
bend and twist rigidities that can become mode-number
dependent. The strength and character of this depen-
dence varies with bundle architecture, such as the ar-
rangement of filaments in the cross section and pretwist.
Two paradigmatic cases of bundle architecture have
been discussed (see Fig. 2): the first assumes filaments to
be arrangedhomogeneously throughout the cross-section,
for example on a square or triangular lattice. This ge-
ometry is particularly relevant for F-actin bundles. The
second architecture has the filaments arranged on the
surface of a cylinder as is the case for microtubules. For
all bundle architectures, the bending rigidity depends on
mode-number qn as κB(n) ∼ k×q−2
dominated regime, as the bending rigidity is proportional
to the shear stiffness, k×, of the crosslinks [11]. It is
in this parameter regime that the bundle behaves qual-
itatively different than either a homogeneous beam (ob-
tained in the ”fully coupled” limit) or an assembly of
”decoupled” filaments. Each architecture has its own
universal ratio of maximal to minimal bending rigidity,
independent of the specific type of crosslink induced fil-
ament coupling. For microtubules (without pretwist) we
find the ratio r = 1+2/sin2(π/N), which is in reasonable
agreement with the available experimental data.
An important factor in determining the strength of
crosslink-induced filament coupling is the pretwist (he-
licity) of the bundle. Numerical computation shows that
the effective bending rigidity decreases with increasing
the pretwist. This has interesting consequences for mi-
crotubules, where the amount of pretwist depends on
the number of protofilaments, N. Different microtubule
types are therefore predicted to have different variations
in their effective bending rigidity.
could be tested in experiments that are able to select
the microtubule type and measure their bending rigidity
independently.
We have discussed several further observables, static
and dynamic, that could be relevant to experiments. We
have shown that the concept of the persistence length
becomes ambiguous and depends on the observable used.
The usual definition via the tangent-tangent correlation
function is shown to lead to a persistence-length that
depends on the scale of observation. Further observables
that are effected by the mode-dependent bending rigidity
are the force-extension relation or the buckling force [12].
Interestingly, in the intermediate regime the latter is con-
stant and independent of bundle length.
The dynamic properties of bundles are characterized
by a complex cross-over scenario which is in one-to-one
correspondence with the three regimes of decoupled, in-
termediate and fully-coupled bending. While decoupled
n. This is the shear
These predictions

Page 12

12
and fully-coupled bending display the usual t3/4in the
correlation function, it is shown that the shear-coupling
leads to an intermediate asymptotic regime, where the
correlations only grow as t1/2. The response functions
for longitudinal and transverse forces also reflect these
different regimes. In contrast to the WLC, they are not
just proportional to each other but show distinct fre-
quency dependences. These findings may be relevant for
microrheological experiments, with imbedded bead par-
ticles directly coupling to the transverse bundle fluctua-
tions.
Possibilities for future studies may be to look into the
effects of filament fracture or lattice defects. The elas-
tic energy represents a harmonic approximation which
should be extended to include nonlinear effects. Espe-
cially for microtubules one may expected nonlinear ef-
fects to play an important role in bundle mechanics. For
example, it may be important to consider that protofil-
aments in their unstressed state are not straight but
bend radially outwards. Aditional complications could
also arise from the fact that some microtubules are not
transversly isotropic as we have assumed here, but have a
”seam”, where protofilaments are offset relative to their
neighbors. Failure modes under axial compressive forces
have been discussed in a model for microtubules that
starts from a transversly isotropic shell theory [58]. It
would be interesting to compare the results of a gener-
alized WLB model – to include cross-sectional deforma-
tions – with their results for the critical buckling forces.
One mode of failure, the ovalization of the microtubule
cross-section (Brazier effect) may for example be taken
into account by adjusting the cross-sectional coordinates
with the help of an “ovalization parameter”. This would
then have to be determined together with the other de-
grees of freedom from the equilibrium equations.
Acknowledgments
The authors acknowledge fruitful discussions with An-
dreas Bausch, Mark Bathe, Mireille Claessens and Karen
Winkler. CH acknowledges the von-Humboldt Feodor-
Lynen, the Marie-Curie Eurosim and the ANR Syscom
program for financial support.
Deutsche Forschungsgemeinschaft for support through
Grant Fr 850/8-1, and to the German Excellence Ini-
tiative via the NIM program. We also acknowledge the
hospitality of the Aspen Center for Physics where part
of this work was completed.
EF is grateful to the
APPENDIX A: EXACT SOLUTION FOR
BENDING AND TWIST RIGIDITIES
In this appendix the effective bending and twist rigidi-
ties are calculated exactly from the WLB Hamiltonian,
Eqs. (1) - (4). To this end the u-variables have to be
integrated over to define an effective WLC Hamiltonian.
First, we have to show that there are no terms in the
shear Hamiltonian, Eq. (3), that would couple the dif-
ferent Euler angles θ,φ,ψ. To this end we use Eq. (4)
in Eq. (3) and specialize to a square cross-section. The
resulting shear Hamiltonian can then be written as
Hshear =
k×
2δ
?
s
?
ij
?
(uij− ui+1,j− bψ′zj+ bφ)2
+ (uij− ui,j+1+ bψ′yi− bθ)2?
where yi = b · (i +1
filament is labeled by the pair of indices (i,j), which de-
notes its location in the ith row and the jth column of
the square cross-section. The only terms that couple the
different Euler angles are b2ψ′φ?
rangement of the filaments,?
space. For the transformation we use cos-modes, which
are appropriate for bundles with pinned boundary condi-
tions. Writing the u-dependent part of the Hamiltonian
in matrix form βH =1
2
?
??
?
2
kl
,(A1)
2) and zj = b · (j +1
2). Here, each
ijzjand b2ψ′θ?
jzj= 0 and?
ijyi.
These identically vanish because of the symmetric ar-
iyi= 0.
The remaining calculations are performed in Fourier-
klukAklul+?
lulblone needs
the following formula valid for a Gaussian integral
k
dukexp
?
−1
2
?
?
kl
ukAklul−
?
?
l
ulbl
?
(A2)
= exp−1
¯ ukAkl¯ ul−
l
¯ ulbl
?
,
where ¯ u is obtained from
∂(βH)/∂uk= 0. (A3)
Having found the solution ¯ u, we can finally bring
Eq. (A2) in the form of Eq. (8) to read off the effective
bend and twist rigidities.
1.Bending
Here we solve Eq. (A3) for the case of bending of the
square bundle. It proves useful to introduce the dimen-
sionless crosslink shear stiffness α = k×L2/ksδ2. We can
then write Eq. (A3) as
(qL)2uk− α∂2uk= 0. (A4)
We also defined the discrete second derivative ∂2uk =
uk+1−2uk+uk−1. Note, that we are working in Fourier-
space so all quantities should carry an additional sub-
script relating to the mode-number n. As different modes
don’t mix, there is no harm in dropping it for the mo-
ment.
At the outer edges of the bundle k = −M,M − 1 we
find
uM−1− uM−2=(qL)2
α
uM−2+ bθ,(A5)

Page 13

13
and
u−M+1− u−M=(qL)2
α
u−M+ bθ.(A6)
The Eq. (A4) is solved by uk= Amk
m± =
2
(qL)2/α. The constants A,B are adjusted to fulfil the
boundary conditions Eqs. (A5) and (A6). This solution
for the axial stretching variable ukis plotted in Fig. 6.
To obtain the approximated bending rigidity of
Eq. (10) one first has to insert the assumption uk =
∆u · (k + 1/2) into the Hamiltonian and then minimize
with respect to the single variable ∆u.
This way one finds
++ Bmk
−, where
1
?x ±√x2− 4?
and we have defined x = 2 +
∆u =
−bθ
6α(√M + 1) 1 +
√M
, (A7)
which has to be reinserted into the Hamiltonian to yield
Eq. (10).
2.Twist
In the case of pure twisting the same analysis can be
done to calculate the effective twist rigidity. The differ-
ence to the bending case is that now the full 2d boundary
value problem has to be solved. The equation determin-
ing the axial stretching uij of the filament indexed by
(i,j) is
(qL)2uij− α∆uij= 0,(A8)
where the operator ∆ = ∂2
version of the discrete Laplacian encountered above. The
finite difference operator ∂iis defined as ∂2
ui−1j− 2uij. As for the case of pure bending, here, the
twist variable ψ enters only via the boundary terms.
The classic theory to calculate the twist rigidity of
beams has been given by Saint-Venant [27].
approach the axial displacements are found by solving
Laplace’s equation ∆u(y,z) = 0 on the appropriate do-
main of the cross-section. We see that Eq. (A8) reduces
to the Laplace equation in the limit α ∼ k×→ ∞, which
is the reason why in Fig. 9 the continuum limit is ap-
proached with increasing shear stiffness at fixed bundle
size. The remaining difference stemming from the dis-
cretization into N filaments represents only a small effect.
In Saint-Venant theory it is well known that for rectan-
gular domains two types of solutions appear, depending
on the aspect-ratio of the cross-section. We illustrate
the different symmetry properties of these solutions in
Fig. 11.
i+ ∂2
jis the two-dimensional
iuij= ui+1j+
In this
10203040
10
20
30
40
510 15202530
10
20
30
40
FIG. 11: Illustration of the two types of solution obtained for
the stretching uij of filament (i,j). Left: square cross-section
of 40 ⋆ 40 filaments. Right: 30 ⋆ 40 filaments
APPENDIX B: RELATION BETWEEN THE Ωα,k
OF FILAMENT k AND THE Ωα OF THE
CENTRAL LINE
This appendix gives some details on the description of
the bundle kinematic degrees of freedom. The goal is to
relate the generalized curvatures Ωα,k of filament k, to
the Ωαof the bundle central line. That these need not
necessarily coincide is best illustrated by discussing an
example. If the central line is twisted but not bent, Ω3
remains as the only non-vanishing component of the cur-
vature vector. The filaments themselves, however, twist
and bend as they trace out a helical path with radius R.
Their bending energy is κfL(RΩ2
Ω3.
As a second example consider the bending of the cen-
tral line, Ω1= 1/R, where R is the radius of curvature.
The filaments that lie at a distance b away from the cen-
tral line naturally have a different radius of curvature,
R ± b, and thus a different bending energy. The correc-
tion is again of higher order and scales as b2Ω4
out, that all effects like the two just mentioned only con-
tribute to higher order. To lowest order we will find that
Ωα= Ωα,k.
We define the vector rk(s) = r0(s) + Rk(s) to point
to filament k at arclength position s. The central line
of the bundle is thereby given by r0(s). The position
of each filament in the cross-section is parametrized by
Rk(s) = Akd1(s) + Bkd2(s), where Akand Bkare con-
stants independent of arclength s and deformation of the
bundle. This, in particular, implies that in the reference
state the filaments are always straight and untwisted. We
also need the derivative of Rkwith respect to s,
3)2, of fourth order in
1. It turns
R′
k= (BkΩ1− AkΩ2)t + Ω3R⊥
where we have used the Frenet-Seret equations and de-
fined R⊥
to filament k is given by
k
(B1)
k= Bkd1(s) − Akd2(s). With this the tangent
tk=
1
N((1 + BkΩ1− AkΩ2)t + Ω3R⊥
with an appropriate normalization factor, N. One finds,
that the filament tangent is parallel to the central-line
k),(B2)

Page 14

14
only if the twist vanishes, Ω3 = 0.
twist, the filament tangent is rotated around the vector
Rk relative to the central tangent t. The magnitude of
the rotation is Ω3Rkand depends on the distance Rk=
|Rk| = |R⊥
In order to derive expressions for the remaining two
unit vectors d1,kand d2,klet us assume, for the time be-
ing, that no other rotation is allowed that may re-orient
the local frame of filament k relative to the central frame.
With this assumption filaments are not allowed to twist
relative to their neighbors as this would correspond to a
rotation around tk. In the bundles we consider, this in-
ternal twist motion can safely be neglected as crosslinks
provide for permanent rigid inter-filament connections.
As explained below, these internal twist modes may nev-
ertheless be important at the time of bundle formation.
The local frame can then be related to the central
frame by
For finite bundle
k| of the filament to the central line.
d1,k = d1− BkΩ3sinαt + O(bkΩ3)2
d2,k = d2+ AkΩ3cosαt + O(bkΩ3)2
(B3)
(B4)
One can now express the generalized curvatures of the
filaments in terms of the Ωαof the central line. For sim-
plicity we here consider the case of pure twisting where
Ω1= Ω2= 0. One then finds
Ω3,k ≡ (∂skd1,k) · d2,k
= Ω3− Ω3Ω′
(B5)
3AkBk= Ω3(1 + O(b2Ω3
L
))
Note, that the difference in length between the central
line and the individual filaments should be taken into ac-
count in the parametrization in terms of arclength, how-
ever, only contributes to higher orders. More formally,
∂s/∂sk= 1 + O(bΩ).
In addition to twist, filaments also bend
Ω1,k ≡ (∂skd2,k) · d3,k
= Ω′
(B6)
3Ak+ Ω2
3Bk
(B7)
and
Ω2,k ≡ (∂skd1,k) · d3,k
= −Ω′
(B8)
3Bk+ Ω2
3Ak.(B9)
which reduces to the well known expression for the cur-
vature of a helix if Ω′
Finally, let us comment on what would happen if we
did allow the filaments to twist individually, and relative
to their neighbors. This filament twist can be described
by an additional rotation, ψk(s) of the local unit vectors,
d1,k,d2,karound the tangent tk.
The new vectors are then given by
3= 0.
e1,k = cosψkd1,k+ sinψkd2,k
e2,k = −sinψkd1,k+ cosψkd2,k,
and the bundle twist, Ω3, can be calculated as before
(B10)
(B11)
Ω3,k= Ω3+ ψ′
k.(B12)
The twist energy should thus be written as
Htwist =
κt
2
?
s
?
?
k
(Ω3+ ψ′
k)2
(B13)
=
Nκt
2
s
(Ω3+¯ψ′)2+ N(¯ψ′2−¯ψ′2),
where we defined the moments of the filament twist dis-
tribution¯ψ′r=?
nal twist can be assumed to be quenched at the time
of bundle formation. In this case we need not treat the
ψk as dynamical variables for the discussion of bundle
deformation. A finite¯ψ′nevertheless gives the bundle
a certain helicity and imposes pretwist, as discussed in
Section V.
kψ′
k
r/N.
As explained above, in crosslinked bundles this inter-
APPENDIX C: DERIVATION OF SHEAR
DEFORMATION
To calculate the geometric part ∆ of the shear defor-
mation, an expression for the arclength mismatch be-
tween the two points on the filament pair is needed. The
general expression is
∆lk= ∆s +
?s
|t + R′
l| −
?s
|t + R′
k|,(C1)
where the first contribution, ∆s, derives from the possi-
bility that the two points do not correspond to the same
point, s, on the bundle central line.
By using Eqs. (B1), (B2) and expanding one finds
∆ = ∆s +
?s
0
?
b′
lk· t −1
2(R′2
l− R′2
k)
?
,(C2)
where, we defined blk= Rl−Rkpointing from filament
k to filament l. As in the formulation of H0, only the
leading order terms have been accounted for. The (b′
term only contains bending deformations. This may be
seen by setting t = ˆ ex = const appropriate for a pure
twisting of the central line. Then ∆lk = blk· ˆ ex = 0
as the vector blk lies within the cross-section, that is
perpendicular to the tangent.
The last term corresponds to the arclength-difference
acquired between two filaments at different distance to
the central-line (Rk?= Rl). It is clear that the filament
farther out has to go a longer distance, so its crosslinking
sites will stay back in comparison to that of its neighbor
closer in the center of the bundle. However, twist also
produces shear deformation between filaments which lie
at equal distance to the central line (Rl = Rk). This
is embodied in the extra term ∆s, which we treat now
(also see Fig. 4). To derive an expression for the shear
displacement in this case, assume that the two filaments
lie, separated by a distance blk= |blk|, on the surface of
a cylinder of radius R. Twisting the cylinder makes the
filaments wind around it, each taking an angle Ω3R to the
lk·t)-

Page 15

15
cylinder axis. The shear deformation then simply is ∆s =
Ω3Rblk. For arbitrary orientation of the filament pair we
have to write ∆s = tlk·blk, where tlkis the tangent to the
midline between the filament pair. In agreement to what
has been said above, this mechanism does not contribute
to the shear displacement when the filament pair (l,k) is
connected by crosslinks in radial direction. In this case,
tlk⊥ blk, and the shear deformation ∆s = 0.
[1] A. Mogilner and B. Rubinstein, Biophys. J. 89, 782
(2005).
[2] E. Atilgan, D. Wirtz, and S. X. Sun, Biophys. J. 90, 65
(2006).
[3] D. Vignejvic, S. Kojima, Y. Aratyn, O. Danciu, T. Svitk-
ina, and G. G. Borisy, J. Cell Biol. 90, 65 (2006).
[4] A. J. Hudspeth and D. P. Corey, Proc. Natl. Acad. Sci.
USA 74, 2407 (1977).
[5] M. F. Schmid, M. B. Sherman, P. Matsudaira, and
W. Chiu, Nature 431, 104 (2004).
[6] J. Howard, Cellular and Molecular Bioengineering 1, 24
(2008).
[7] M. M. A. E. Claessens, M. Bathe, E. Frey, and A. R.
Bausch, Nature Mat. 5, 748 (2006).
[8] O. Pelletier, E. Pokidysheva, L. S. Hirst, N. Bouxsein,
Y. Li, and C. R. Safinya, Phys. Rev. Lett. 91, 148102
(2003).
[9] O. Lieleg, M. Claessens, C. Heussinger, E. Frey, and A. R.
Bausch, Phys. Rev. Lett. 99, 088102 (2007).
[10] K. M. Schmoller, O. Lieleg, and A. R. Bausch, Phys. Rev.
Lett. 101, 118102 (2008).
[11] M. Bathe, C. Heussinger, M. M. A. E. Claessens, A. R.
Bausch, and E. Frey, Biopys. J. 94, 2955 (2008).
[12] C. Heussinger, M. Bathe, and E. Frey, Phys. Rev. Lett.
99, 048101 (2007).
[13] O. Kratky and G. Porod, Rec. Trav. Chim. Pays-Bas. 68,
1106 (1949).
[14] N. Saitˆ o, K. Takahashi, and Y. Yunoki, J. Phys. Soc. J.
22, 219 (1967).
[15] N. S. Gov, Phys. Rev. E 78, 011916 (2008).
[16] G. M. Guild, P. S. Connelly, L. Ruggiero, K. A. Vranich,
and L. G. Tilney, J. Cell Biol. 162, 1069 (2003).
[17] M. M. A. E. Claessens, C. Semmrich, L. Ramos, and
A. R. Bausch, Proc. Natl. Acad. Sci. USA 105, 8819
(2008).
[18] K. R. Purdy, J. R. Bartles, and G. C. L. Wong, Phys.
Rev. Lett. 98, 058105 (2007).
[19] A. Hilfinger and F. J¨ ulicher, Phys. Biol. 5, 016003 (2008).
[20] K. Alim and E. Frey, Eur. Phys. J. E 24, 185 (2007).
[21] H. Goldstein, C. Poole, and J. Safko, Classical Mechanics
(Addison Wesley, San Francisco, 2002), 3rd ed.
[22] R. Everaers, R. Bundschuh, and K. Kremer, Europhys.
Lett. 29, 263 (1995).
[23] B. Mergell, M. R. Ejtehadi, and R. Everaers, Phys. Rev.
E 66, 011903 (2002).
[24] R. Golestanian and T. B. Liverpool, Phys. Rev. E 62,
5488 (2000).
[25] H. Zhou, Y. Zhang, and Z. can Ou-Yang, Phys. Rev. E
62, 1045 (2000).
[26] H. Mohrbach and I. M. Kulic, Phys. Rev. Lett. 99,
218102 (2007).
[27] A. E. H. Love, The Mathematical Theory of Elasticity
(Dover, New York, 1944), 4th ed.
[28] C. R. Calladine, Theory of shell structures (Cambridge
University Press, 1983).
[29] L. D. Landau and E. M. Lifschitz, Theory of Elasticity
(Pergamon Press, Oxford, 1970).
[30] J. M. Gere and S. P. Timoshenko, Mechanics of Materials
(Nelson Thornes Ltd, Cheltenham, 2002), 5th ed.
[31] A. Kis, S. Kasas, B. Babi´ c, A. J. Kulk, W. Benoˆ ıt,
G. A. D. Briggs, C. Sch¨ onenberger, S. Catsicas, and
L. Forr´ o, Phys. Rev. Lett. 89, 248101 (2002).
[32] F. Pampaloni, G. Lattanzi, A. Jon´ as, T. Surrey, E. Frey,
and E.-L. Florin, Proc. Natl. Acad. Sci. USA 103, 10248
(2006).
[33] A. Kis, G. Cs´ anyi, J.-P. Salvetat, T.-N. Lee, E. Couteau,
A. J. Kulik, W. Benoit, J. Brugger, and L. Forr´ o, Nature
Mat. 3, 153 (2004).
[34] D. Nelson and L. Peliti, J. Physique 48, 1085 (1987).
[35] D. Nelson, T. Piran, and S. Weinberg, eds., Statistical
Mechanics of Membranes and Surfaces (World Scientific,
2004), 2nd ed.
[36] E. Frey and D. R. Nelson, J. Phys. 1 1, 1715 (1991).
[37] K. Kroy and E. Frey, Phys. Rev. E 55, 3092 (1997).
[38] H. Isambert and A. C. Maggs, Macromolecules 29, 1036
(1996).
[39] F. Gittes and F. C. MacKintosh, Phys. Rev. E 58, R1241
(1998).
[40] D. C. Morse, Phys. Rev. E 58, R1237 (1998).
[41] D. C. Morse, Macromolecules 31, 7044 (1998).
[42] G. H. Koenderink, M. Atakhorrami, F. C. MacKintosh,
and C. F. Schmidt, Phys. Rev. Lett. 96, 138307 (2006).
[43] R. Granek, J. Physique II 7, 1761 (1997).
[44] E. Frey, K. Kroy, and J. Wilhelm, The Wiley Polymer
Networks Group Review (John Wiley, 2000), vol. 2.
[45] K. Kroy and J. Glaser, New J. Phys. 9, 416 (2007).
[46] D. Chr´ etien, F. Metoz, F. Verde, E. Karsenti, and R. H.
Wade, J. Cell Bio. 117, 1031 (1992).
[47] D. Chr´ etien and R. H. Wade, Biol. Cell 107, 161 (1991).
[48] M. Kurachi, M. Hoshi, and H. Tashiro, Cell Motil. Cy-
toskeleton 30, 221 (1995).
[49] M. Kikumoto, M. Kurachi, V. Tosa, and H. Tashiro, Bio-
phys. J 90, 1687 (2006).
[50] C. P. Brangwynne, G. Koenderink, E. Barry, Z. Dogic,
F. C. MacKintosh, and D. A. Weitz, Biophys. J. 93, 346
(2007).
[51] K. M. Taute, F. Pampaloni, E. Frey, and E.-L. Florin,
Phys. Rev. Lett. 100, 028102 (2008).
[52] M. G. L. van den Heuvel, M. P. de Graaff, and C. Dekker,
Proc. Nat. Acad. Sci. USA 105, 7941 (2008).
[53] M. G. L. van den Heuvel, S. Bolhuis, and C. Dekker,
NanoLett. 7, 3138 (2007).
[54] M. E. Janson and M. Dogterom, Biophys. J. 87, 2723
(2004).
[55] F. Gittes, B. Mickey, J. Nettleton, and J. Howard, J Cell
Biol 120, 923 (1994).
[56] G. Grason and R. Bruinsma, Phys. Rev. Lett. 99, 098101
(2007).
[57] G. M. Grason, Phys. Rev. E 79, 041919 (2009).
[58] C. Y. Wang, C. Q. Ru, and A. Mioduchowski, Phys. Rev.