Dynamics of a stiff biopolymer in an actively contractile background: buckling, stiffening and negative dissipation

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by  Norio Kikuchi

Dynamics of a stiff biopolymer in an actively contractile background: buckling,
stiffening and negative dissipation
Norio Kikuchi1, Allen Ehrlicher2, Daniel Koch3, Josef A. K¨as3, Sriram Ramaswamy1,4 and Madan Rao5,6
1Centre for Condensed Matter Theory, Department of Physics,
Indian Institute of Science, Bangalore 560 012, India
2Translational Medicine, Brigham and Women’s Hospital,
Harvard Medical School, One Blackfan Circle, Karp 6, Boston, MA 02115, USA
3Institute of Soft Matter Physics, Universit¨at Leipzig, Linn´estraße 5, 04103 Leipzig, Germany
4Condensed Matter Theory Unit, Jawaharlal Nehru Center for Advanced Scientfic Research, Bangalore 560 064, India
5Raman Research Institute, C.V. Raman Avenue, Bangalore 560 080, India
9 6National Centre for Biological Sciences (TIFR), Bellary Road, Bangalore 560 065, India
0
0 We present a generic theory for the dynamics of a stiff filament under tension, in an active
2 medium with orientational correlations, such as a microtubule in contractile actin. In sharp

n contrast to the case of a passive medium, we find the filament can stiffen, and possibly oscillate,
a or buckle, depending on the contractile or tensile nature of the activity and the filament-medium
J anchoring interaction. We present experiments on the behaviour of microtubules in the growth

6 cone of a neuron, which provide evidence for these apparently opposing behaviours. We also
2 demonstrateastrongviolationofthefluctuation-dissipation(FD)relationintheeffectivedynamics
ofthefilament,includinganegativeFDratio. Ourapproachisalsoofrelevancetothedynamicsof

] axons,andourmodelequationsbeararemarkableformalsimilarity tothoseinrecentwork[PNAS
ft (2001) 98:14380-14385] on auditory hair cells. Detailed tests of our predictions can be made using
o a single filament in actomyosin extracts or bacterial suspensions.
s
.
t cytoskeleton | active hydrodynamics | microrheology | contractile | tensile | fluctuation-dissipation
a
ratio | buckling| stiffening | oscillations | neuronal growth cone | hair cells |axons
m
- Abbreviations: F-actin, filamentous actin; FD, fluctuation-dissipation
d
n
o
INTRODUCTION
c
[

1 The cytoskeleton[1] is a dense multicomponent mesh-
v work of semiflexible polymers which interact sterically
6 as well as through active [1, 2, 3] processes. While the
2
blending of polymers industrially requires special effort,
1
the active environment of the living cell provides a set-
4
. tinginwhichpolymerswhichdiffersubstantiallyintheir
1
stiffness are naturally mixed and interact. Moreover,ac-
0
9 tive processes such as polymerization and the working FIG. 1: A stiff filament (“microtubule”) embedded in an ac-
0 of molecular motors lead to the generation of stresses tive isotropic medium consisting of oriented fiaments (“F-
: without the external imposition of flow fields. These actin”). The conformations of the microtubule, aligned on
v an average along the x axis, are described by small trans-
Xi two mechanisms combine to yield a rich range of novel verse fluctuations u⊥(x,t). The active medium can either be
physical phenomena. The role of activity in cytoskele-
contractile or tensile. The orientation of F-actin along the
r tal mechanics is receiving increasing attention, as seen microtubule can either be parallel or normal to it.
a
from many recent theoretical and experimental studies
of the rheology of cells and cell extracts [6, 7, 8]. It is
clear in particular [9] that interactions between different
species of filaments are crucial for cell motility, cell di- tile and tensile activity, although only the former ap-
vision, vesicular transport and organelle positioning and plies to actomyosin. Our treatment applies more gen-
integrity. erally to semiflexible polymers under tension in a wide
In this paper, we make a study of the effect of these variety of active media. We describe the medium by
interactions by modeling the dynamics of a stiff fila- the active generalization of liquid-crystal hydrodynam-
ment, which we will call a “microtubule”, immersed in ics[2,3, 4,5,10,11, 12,13,14]. Forthe purposesofthis
an active medium (Fig.1) with orientational degrees of paper, an active medium is one whose constituent parti-
freedom, which we will call “F-actin”. We emphasize cles possess the ability to extract energy from an ambi-
here that the names “microtubule” and “F-actin” are ent nutrient bath and dissipate it, executing some kind
introduced for convenience: we consider both contrac- of systematic motion in the process. This endows each
2
suchparticlewithapermanentuniaxialstress. Theother
centralingredientofthis workis anchoring, onwhichwe
now elaborate. In general, the interfacial energy of a
liquid-crystalline medium at a wall depends on the rela-
tiveorientationnofthemoleculesofthemediumandthe
normal N to the wall. In the simplest cases, it is lowest
for n parallel, or perpendicular, to N. This interaction
is known as anchoring. In the present work, anchoring
enters through the favoured orientation of the F-actin
whenconfrontedwiththesurfaceofthemicrotubule. We
believe thatstericorpair-potentialeffects shouldleadto
theanchoringoftheF-actinnormalorparalleltothemi-
crotubule. The interplay of the types of anchoring and
activity – contractile or tensile – are fundamental to our
theory. Thefilamentinourstudycanbeviewedasaspa-
tially extended probe of the active medium, generalizing
the microrheometry of [6, 7, 8, 15, 16] by simultaneous
access to a wide range of scales. We make contact with
earlierworkonoscillatoryfilaments[17,18],andtestour
FIG. 2: Cytoskeletal polymers in a fixed and fluorescently
results against observations on microtubules in a back-
stained growth cone of an NG108-15 neuronal cell. In the
groundofcontractileactininneuronalgrowthcones(see
peripheral domain actin filaments (red) form dense networks
Fig.2).
in the veil-like lamellipodium and bundles in the spike-like
Here are our main results: (i) In the absence of activ- filopodia. Microtubules (green) emerge from the central do-
ity, the microtubule will always buckle at large anchor- mainintotheperipherywheretheyarebuckled(yellowarrow-
ing strength, regardlessof the type of anchoring. (ii) An heads)inthelamellipodiumbytheactinnetworkorstabilized
active medium, by contrast, can stiffen or buckle the fil- in the filopodial region (blue arrowheads). For illustration
some microtubules are traced by a dotted white line.
ament, depending on the relative signs of activity and
anchoring. Acontractileactivemediumwithparallelan-
choring alwaysstiffens the filament, as does a tensile ac-
tive medium with normal anchoring, while a contractile x axis (Fig.1), with unit tangent vector ˆt = xˆ +
(tensile) medium with normal (parallel) anchoring pro- δˆt 1+O(∂xu⊥)2,∂xu⊥(x,t) whereu⊥(x,t)aresmall
duces buckling if the strength W of the active stresses is ≃
trans(cid:0)verse fluctuations, and (cid:1) y,z, immersed in a d-
large enough (see Fig.3). When the nematic correlation dimensional active medium c⊥ha≡racterized by Q, a sym-
length of the F-actin medium is large compared to the
metric traceless nematic order parameter [19]. The ef-
linear dimension L of the sample transverse to the mi-
fects of contractile or tensile active stresses enter the
crotubule, the buckling wavelength decreases with L as
equations of motion
1/√WL and 1/√WlogL respectively in two and three
dimensions. (iii) Our observations of the conformations 1
ofmicrotubulesinthegrowthconeofneuronsareinqual- ∂tu⊥−v⊥(x,r⊥=0,t) = −γδF/δu⊥+f⊥, (1)
itative accord with these predictions. (iv) Activity leads 1
∂ Q= δF/δQ+η, (2)
to a breakdown of the fluctuation-dissipation relation: t
−ζ
most dramatically, in the regime of strong stiffening, we
predictthattheeffectivedissipationturnsnegativewhen for u⊥ and Q through the hydrodynamicvelocity field v
thefrequencycrossesathreshold. Thisisconsistentwith whose dynamics is governed by Eq.3 below. The Gaus-
the observationsof[17]onauditoryhaircellsand[18]on sian,spatiotemporallywhite noisesf⊥,η inEqs.1,2have
axons,andsuggeststhatanegativeforce-velocityrelation strengths 2N1, 2N2, reducing respectively to 2kBT/γ,
atfinite frequencyshouldbe agenericfeatureofactively 2kBT/ζ for the equilibrium case. In Eqs.1,2, u⊥ and
stiffening systems. Indeed, the phenomenological model Q are coupled only through the free-energy functional
of[17]emergesasalimitingcaseofourfundamentalthe- F[u⊥,Q] = Ff[u⊥] + FLD[Q] + Fanc[u⊥,Q]. The
ory. filament free energy Ff[u⊥] = 0Ldx[(σ/2)(∂xu⊥)2 +
(κ/2)(∂x2u⊥)2] contains bending eRnergy with rigidity κ
and an imposed tension σ. [20, 21] to leading order in
A FILAMENT IN AN ACTIVE MEDIUM ∂xu⊥. The Landau-de Gennes free energy FLD[Q] =
dx d2r⊥[(a/2)Q2+(K/2)( Q)2]describesincipientori-
∇
Consider a stiff, locally inextensible filament of to- RentaRtional ordering in the medium [19]. We work here
tal contour length L, coinciding on average with the inthe isotropicphasewithcorrelationlength K/a; a
∼
p
3
studyofthenematicphasebygeneralising[22]toinclude ACTIVE STIFFENING AND BUCKLING
activitywillbepresentedelsewhere[23]. Thefilamentan-
chorstheorientationaldegreesoffreedomofthemedium Wearenowinapositiontoinvestigatethe(in)stability
through of the filament. At thermal equilibrium (W = 0)
sgn[α]= sgn[β];indeed,αβ = A2/ζγ <0irrespective
Fanc[u⊥,Q]= A2 0Ldxˆt·Q(x,r⊥ =0)·ˆt ofthesign−ofA. Thisimpliesab−ucklinginstabilityofthe
≃const.+A 0Ldx[∂xuR⊥·Qx⊥(x,r⊥ =0)+O(∂xu⊥)2] filelsasmoefnwthaesththeer tahnechaonrcinhgorsintrgenisgtphariasllienlcroeraspeedr,perengdaicrud--
R
with negative and positive A corresponding respectively lar. This is consistent with observations [22] of buckling
to parallel and normal anchoring. Note that Eq.1 gen- of filaments in isotropic solutions of fd virus (see their
eralises [20, 21] to include anchoring and hydrodynamic Fig. 1a). WhenactivityW is switchedon,the signsofα
flow. In Eq.2, we ignore flow-orientation coupling terms and β become independent. For large enough W , α>0
| |
[19] [29]. for contractile (W < 0) activity and α < 0 for tensile
For thin-film samples at large F-actin concentration, (W >0) activity, while β is separately controlled by the
as in the case of the lamellipodium of adhering cells, it nature of the anchoring (Fig.3).
is appropriate to treat the hydrodynamic velocity field Defining the correlation length ξ K/a of the
≡
in a local-friction approximation. We therefore write F-actin medium, we focus on two limitipng cases: (I)
Γvi(x,r⊥,t)= jσij, with Γ µ/ℓ2 where µ is the cy- deep in the isotropic phase; (II) close to the transition
−∇ ∼
toplasmic viscosity, and the screening length [24] ℓ is no to a nematic phase. In (I), q ξ 1, so Eq.6 holds
x
≪
largerthan the film thickness. We ignore pressuregradi- with qω = a/K for small ω. Thus, for a filament in
entsontheassumptionthatthefilmthicknessadjusts to a strongly apctive medium, contractile (tensile) activity
accommodate these. The crucial piece of the stress σ with parallel (normal) anchoring (αβ > 0) leads to en-
ij
is the active contribution σiajct ≃ Wc0Qij(x,r⊥,t) [30] hanced tension, i.e., stiffening. On the other hand, for
[2, 3, 4, 5, 11, 12], where W < 0 and W > 0 respec- strong contractile (tensile) activity with normal (paral-
tively correspond to contractile and tensile stresses, and lel)anchoring(αβ <0)thefilamentbecomesunstableto
c0 is the mean F-actin concentration. To leading order buckling. Thisisexplainedgraphicallyin(Fig.3). In(II),
ingradientsandlinearorderinfilamentundulations,the
velocity transverse to the microtubule is
v⊥(x,r⊥,t)= (Wc0/Γ)∂xQx⊥(x,r⊥,t). (3)
−
FromEqs.1-3,theeffectiveFourier-transformedequation
of motion for u⊥(qx,ω) is
σ κ
(cid:18)−iω+ γqx2+ γqx4(cid:19)u⊥ =iqxαQx⊥(r⊥=0)+f⊥, (4)
where u⊥ is coupled to Q only at r⊥ =0.
FIG. 3: A perturbation of a “microtubule” (black) leads,
iq βζ ζ through anchoring, to a distortion of the “F-actin” medium
Qx⊥(r⊥=0)= x Gqωu⊥+ Gqωη(qx,q⊥,ω), and hence of the active stress profile. Dependingon whether
K Z KZ
q⊥ q⊥ the actin is anchored parallel or normal to the microtubule,
(5)
andwhethertheactivestresses(bluedouble-arrows)arecon-
where q2 ζiω/K+a/K+q2 and G (q2+q2)−1,
and q⊥≡ω≡0Λ−dd−1(q⊥/2π), with uxltravioleqtωc≡utoff⊥Λ ∼ω(fil- tornadcatrilyefloorwtesn(srieledaalrornowgst)heeitahcetirnsufiplapmreesnstos,rtehnehraenscuelttihnegpseecr--
amenRt thRickness)−1. The signs of α (A/γ Wc0/Γ) turbation, leading to active stiffening or buckling.
≡ −
and β A/ζ decide the fate of the filament (stiffening
≡−
or buckling) in the following analysis. Since G
q⊥ qω ∼ qxξ 1 (which should be accessible in F-actin, see [25])
ln(Λ2/qω2)in3dandπ/(2qω)in2d,weobtainRthe disper- so th≫at a wide range of modes of Q contribute in Eq.5.
sion relations for u⊥(qx,ω) for the case of an unbounded In dimension d =3, Eq.6 applies with qω2 ≃ qx2−ζiω/K,
medium: so that the dispersion relation differs from case (I) only
by a logarithmic factor. In d = 2, however, solving the
σ + αβζ ln Λ2/q2 q2+O(q4), d=3
−iω = −hγ−hσγK+ α4βKζ(cid:0)qω−1iωqx2(cid:1)i+xO(qx4), x d=2. (6) resulting cu−biciωeq=uaYti(o|αnβw|/e4fi)2n/d3(ζ/K)1/3qx4/3 (7)

The case of a confined medium is discussed towards the where Y = ( 1/2 i√3/2) for αβ>0 and 1 for αβ<0.
− ±
end of the paper. Thus, in 2d, close to the ordering transition of the F-
4
actin, while stability is determined by the sign of αβ as fluctuations in that degree of freedom, the Fluctuation-
before, the stable stiffening case αβ>0, which can arise Dissipation Theorem (FDT) [26], which applies to sys-
only for an active system, shows a damped oscillatory tems at thermal equilibrium, says C(t) = k Tχ(t),
B
−
response,which should appear as dispersive propagating where k is Boltzmann’s constant and T is the temper-
B
wavesonthefilament,withspeed q1/3. Intheunstable ature. This profound and universal connection can be
x
∝
case,thebucklingwavelengthcanbefoundbycomparing understoodin the familiar contextofa particle undergo-
thenegativetensionandbendingelasticityterms. Foran ing Brownianmotioninafluid, where boththe damping
F-actin medium of finite lateral extent L, scaling argu- of an initially imposed velocity and the random motion
ments applied to Eqs.6,7 lead to a buckling wavelength oftheparticlewhennovelocityisimposedarisefromthe
varying as 1/√WL (d = 2) and 1/√W logL (d = 3) as same microscopic collisions with molecules. For equilib-
claimed earlier in the paper. rium systems, this allows one to obtain transport quan-
The ideas presented above are best tested in a model tities suchasconductivitywithoutdrawingacurrent. In
systeminwhichbothparallelandnormalalignmentofF- systemsfarfromequilibrium,butwherethebathproduc-
actinonmicrotubulesarenaturallyrealised. Thegrowth ingthefluctuationsisstillthermal,ajudiciousdefinition
cones of neurons provide a convenient system for this ofvariables[27]canresurrecttheFDT.Formoregeneral
purpose, and allow a demonstration of activity-induced nonequilibrium systems too, a comparison of correlation
stiffening as well as buckling. Our observations on this and response can sometimes offer a useful notion [28] of
system are qualitative but in clear accord with the pre- an effective temperature. Our model system shows radi-
dictions of this paper. As seen in Fig.2 microtubules cal departures from such benign behaviour.
are stronglybuckled in the wide centralregionknownas We restrict attention to parameterranges where there
the lamellipodium and show a stabilized straight shape is no instability, so a steady state exists. Let S(qx,ω)
≡
in the spiky protrusions called the filopodia. As we have x,texp(iqxx−iωt)hu⊥(x,t)·u⊥(0,0)i be the correlation
remarked,purelyequilibriumeffectsinmostsystemsgen- Rfunctionandχ′′(q ,ω)theimaginarypartoftheresponse
x
erally lead to buckling. It seems, however, that the dy- to an external force h(x,t) coupled to u⊥ via a term
namic activity within the lamella may contribute to mi- dxh u⊥ inthefree-energyfunctionalF inEq.1. The
− ·
crotubule stiffening in the periphery, as well as buckling deRparture from unity of the fluctuation-dissipation (FD)
in the central region. We suggest the following mecha- ratioR(q ,ω) (ω/2k T)S(q ,ω)/χ′′(q ,ω)isaquanti-
x B x x
≡
nism : when the growing microtubule enters the lamel- tativemeasureofnonequilibriumbehaviour. Wefindnot
lipodium, it meets the actomyosin network, with actin only that R(q ,ω) depends on its arguments but that it
x
filaments pointing in a variety of directions. The typical can turn negative for the stable stiffening case αβ >0.
encounter will have a substantial angle between micro- The calculation, from Eqs.4,5, is straightforward. We
tubuleandactinfilament. Theanalysisabovewouldthen find
predict that the microtubule should buckle, as it does.
Uponenteringafilopodialregion,themicrotubulemeets χ (q ,ω)=ω 1−αβ(ζ/K)2qx2Σ(qω) , (8)
x
actin aligned parallel to it and is stiffened, which again ′′ (cid:20) Dqxω (cid:21)
is consistent with our mechanism. However, whether
tohnecroenltervaacnttiliatyc,tivoirtywhisetehnetrirseplyecniaolniz-esdpepcrifiocteiannsdabreasiend- wherqe2+a/KD+qqx2ω2+(ζω/Kan)d2 −1are Σs(tqrωi)ctly positiv=e
q⊥ ⊥ x
volved remains an open question. In another relevant aRnd(cid:2)(cid:0)even in ω, an(cid:1)d the FD ra(cid:3)tio
study, Brangwynne et al. [8] observed significant bend-
ing fluctuations of microtubules in reconstituted acto- R = N1γ 1+ α(αN2/N1+β)(ζ/K)2qx2Σ(qω) .
myosin networks (see Fig.1 in [8]). Since the actomyosin qxω k T (cid:20) 1 αβ(ζ/K)2q2Σ(q ) (cid:21)
B − x ω
iscontractile,ourtheorywouldpredictthatthecontacts (9)
betweenactinandmicrotubule areatnear-normalalign- At thermal equilibrium (W =0) N1 = kBT/γ, N2 =
ment, and we look forward to independent tests of this. kBT/ζ, α = A/γ, and β = A/ζ, so that αN2/N1 =
−
Indeed, our work provides a theoretical justification for β. Hence, the second term in the bracket in Eq.9
−
the pointlike normal force that [8] suggest is responsible vanishes and the fluctuation-dissipation ratio becomes
for the buckling. unity as expected. With activity the ratio becomes a
strong function of frequency and wavenumber, through
the quantity Σ(q ). In the stiffening case αβ = (A/γ
ω
FLUCTUATION-DISSIPATION RATIO AND Wc0/Γ)( A/ζ) WA>0, if the strength W of activit−y
NEGATIVE DISSIPATION − ∼
is large enough, as ω crosses a threshold which depends
on q and W, we see from Eqs.8,9, χ can pass through
x
′′
If χ(t) is the displacement of a degree of freedom of zero, and hence R can diverge. Past this threshold,
qxω
a system at time t in response to an impulsive force at both turn negative. Thus one obtains a giant or even a
time 0, and C(t) is the time-correlation of spontaneous negative FD ratio as a result of suppressed dissipation
5
rather than enhanced noise. The FD ratio Eq.9 is some- broad accord with our theory. We look forward to more
timesreferredto[17]astheratioofan“effectivetemper- quantitativetests,onfilamentssuspendedincellextracts
ature” T to the thermodynamic temperature T, and as well as suspensions of swimming organisms, to see if
eff
a negativeT inthis sensehas beenobservedinexper- the scenarii we propose are observed.
eff
iments on hair cells [17]. Our theory Eqs.1-3, rooted in NKwassupportedbytheIIScCentenaryPostdoctoral
the active-hydrodynamic approach of [2, 3, 4, 5, 11, 12], Fellowship. SR acknowledges support from the Depart-
provides a fundamental basis for the model presented in ment of Science and Technology,India through the Cen-
[17] and suggeststhat the phenomenon ofnegative dissi- tre for Condensed Matter Theory and DST Math-Bio
pation should be widely observed in active systems. To Centre grant SR/S4/MS:419/07. MR thanks the HFSP,
apply our treatment directly to axons [18] and hair cells and SR and MR thank CEFIPRA project 3504-2. We
[17],weconsiderafilamentconfinedtoatubeofradiusL thankP.A.Pullarkat,J.-F.Joanny,J.Prost,N.Uchida,
in the directions and unbounded along x, and assume H.OgawaandT.Kawakatsuforstimulating discussions.
⊥
the correlation length ξ K/a of the active medium
≡
is of order L. The integralspoverq⊥ in Eq.5 are replaced
by sums dominated by a single mode with wavenumber
of order1/L. This yields an effective equationof motion
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CONCLUSION
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