Motor-Driven Bacterial Flagella and Buckling Instabilities

4.4 MB·

by  Reinhard Vogel

EPJ manuscript No.
(will be inserted by the editor)
Motor-Driven Bacterial Flagella and Buckling Instabilities
Reinhard Vogel1 and Holger Stark1
1 Institute for Theoretical Physics , TU Berlin
2 Institute for Theoretical Physics , TU Berlin
Received: date / Revised version: date
2
1 Abstract. Many types of bacteria swim by rotating a bundle of helical filaments also called flagella. Each
0 filamentisdrivenbyarotarymotorandaveryflexiblehooktransmitsthemotortorquetothefilament.We
2 model it by discretizing Kirchhoff’s elastic-rod theory and develop a coarse-grained approach for driving
n thehelicalfilamentbyamotortorque.Arotatingflagellumgeneratesathrustforce,whichpushesthecell
a bodyforwardandwhichincreaseswiththemotortorque.Wefixtherotatingflagelluminspaceandshow
J thatitbucklesunderthethrustforceatacriticalmotortorque.Bucklingbecomesvisibleasasupercritical
3 Hopf bifurcation in the thrust force. A second buckling transition occurs at an even higher motor torque.
We attach the flagellum to a spherical cell body and also observe the first buckling transition during

] locomotion. By changing the size of the cell body, we vary the necessary thrust force and thereby obtain
h
a characteristic relation between the critical thrust force and motor torque. We present a sophisticated
p
analytical model for the buckling transition based on a helical rod which quantitatively reproduces the
-
o critical force-torque relation. Real values for motor torque, cell body size, and the geometry of the helical
i filamentsuggestthatbucklingshouldoccurinsinglebacterialflagella.Wealsofindthattheorientationof
b
pulling flagella along the driving torque is not stable and comment on the biological relevance for marine
.
s bacteria.
c
i
s PACS. PACS-key discribing text of that key – PACS-key discribing text of that key
y
h
p 1 Introduction thermore, each flagellum is reported to be relatively rigid
[
and visible deformations due to rotation are not reported
Many bacteria such as Escherichia coli and Salmonella
1 [5].Ingeneral,videosshowacomplexbehaviorofasingle
v typhimurium swim by rotating a bundle of helical flagella flagellum when it interacts with other flagella, with the
8 [1].Nature’ssimpleandingenioussolutionforlocomotion wall, or when it goes through different polymorphic con-
2 at low Reynolds number has already inspired researchers
formations [6]. In this context, recent articles study the
6 to apply rotating flagella to perform such diverse tasks as
synchronization and bundling of two or more flagella due
0 pumping fluid [2] or manufacturing nanotubes [3]. Even
to hydrodynamic interactions [7,8,9,10].
.
1 artificial helical flagella already exist [4].
Thepolymorphismoftheflagellumisafascinatingand
0 The flagellum in the bundle consists of three parts;
intensively studied subject [11,12,13,14,15,16,17,18,19].
2 the rotary motor, a short and very flexible proximal hook
1 that couples the motor to the third part, the long heli- Using a coarse-grained molecular model, we recently ad-
v: cal filament [1,6,5]. The motors are embedded at differ- dressed the question why the normal polymorphic state
is realized in a flagellum [20] and also developed a sim-
i ent locations of the cell wall so that the flagella have to
X ple model for flagellar growth [21]. Based on an extended
bendaroundthecellbodytoformabundle.Bacteriasuch
r as Escherichia coli and Salmonella typhimurium use this Kirchhoff theory for the helical filament, we modeled the
a polymorphism of the flagellum [22] and were able to re-
bundletoperformarun-and-tumblemotionwhichenables
produceexperimentalforce-extensioncurveswhereapoly-
them to follow a chemical gradient (chemotaxis). After
morphic transition is induced by an external force [18].
swimming for about 1s, the sense of rotation of one mo-
tor reverses and the attached flagellum leaves the bundle. In this article we concentrate on the normal form of
It goes through a sequence of polymorphic conformations a single bacterial flagellum, model it by the discretized
untilthemotorreversesitsrotationaldirectionagain.The version of Kirchhoff’s elastic-rod theory and develop a
flagellum returns to its original or normal helical form coarse-grained approach for driving the helical filament
andrejoinsthebundle.Duringthistumblingevent,which by a motor torque. A rotating helical flagellum produces
lasts for about 0.1s, the bacterium changes its swimming a thrust force as explained in fig. 1 that adds up along
direction randomly. It is interesting that flagella of cells, the filament and then pushes the cell body forward. We
which are clued to a surface, do not bundle [5,6]. Fur- reporttwobucklinginstabilitiesofafixedhelicalfilament
2 Reinhard Vogel, Holger Stark: Motor-Driven Bacterial Flagella and Buckling Instabilities
andtheappliedtorqueisbalancedbyfrictionalforcesand
torques continuously distributed along the filament. Wol-
gemuth et.al. investigated a rod with one clamped and
one free end rotating around its axis. They observed two
regimes separated by a supercritical (i.e. continues) Hopf
bifurcation.Whentherotationalfrequencyexceedsacrit-
ical value, the straight filament starts to bend and per-
forms a whirling motion [24]. In Brownian dynamics sim-
Fig. 1. Each segment of a rotating helical flagellum experi-
ulations Wada and Netz observed for the same conditions
ences a frictional force F that is not antiparallel to the local
a subcritical (i.e. discontinuous) Hopf bifurcation where
velocityvduetotheanisotropicfrictionofarod.Whereasthe
force component perpendicular to the helix axis averages to the strongly bent filament nearly folds back on itself [25].
zero over one helical turn, the parallel component adds up to On the other hand, a rod tilted with respect to the rota-
the thrust force. For a detailed treatment see appendix A. tional axis bends slightly due to friction at low rotational
velocity. At a critical value, a discontinuous transition to
a helical rod shape occurs [26].
(a) (b)
Inthisarticlewetreatbucklinginstabilitiesforthebi-
buckled
ologically relevant helical filament. The problem is more
complexduetothecharacteristicrotation-translationcou-
plingandthefactthatwedonotfixtheorientationofthe
helical filament. The content of the article is organized as
straight
follows. In sect. 2 we explain how we model the motor-
driven bacterial flagellum and how we perform the sim-
ulations. In sect. 3 we present and discuss our numerical
resultsforbothbucklinginstabilitiesforthefixedfilament
andthenaddressthefirstbucklinginstabilityduringloco-
motion of the filament. In sect. 4 we formulate a buckling
Fig. 2. a)Anelasticrodbucklesundertheinfluenceofacom- theory for a helical rod and show that it quantitatively
pressional force F and an external torque M. b) The critical
reproduces the critical force-torque relation in the bio-
valuesF andM atwhichbucklingoccursobeyacharacteris-
c c logically relevant regime. We close with a summary and
ticrelation.Thegraphdepictsrelation(1)validforarodwith
conclusions.
fixed ends.
forincreasingmotortorque.Thefirstinstabilityoccursin 2 Modeling the motor-driven bacterial
the biologically relevant regime. The straight helical fil-
flagellum
ament starts to bend under the influence of the acting
thrust force similar to a rod that buckles under its own
We start with a short review of the elasticity model and
load. The buckling instability is visible as a supercritical
the dynamics of a helical filament in sects. 2.1 and 2.2
Hopf bifurcation in the thrust force. It also occurs when
following our previous work [22] and then explain how we
the filament is allowed to move by attaching it to a load
model the motor-driven hook in sect. 2.3. A summary of
particle. We will develop an analytical model based on a
the simulation parameters follows in sect. 2.4.
rigid helical rod that explains the buckling transition and
reproducesquantitativelythecriticalforce-torquerelation
from our simulations.
An elastic rod buckles under the influence of a com- 2.1 Elasticity model of a helical filament
pressionalforceandanexternaltorqueactingatbothends
oftherod[fig.2(a)].Thisisoneofthefirstexamplesfora We describe the conformation of the helical filament with
bifurcation and Euler was the first to provide the theory contourlengthLbythespacecurveofitscenterliner(s),
for the critical load force at zero torque. In general, crit- where s is the arc length. In addition, we attach a mate-
ical force and torque for a rod with fixed ends obey the rial frame of three orthogonal unit vectors {e ,e ,e }, to
1 2 3
relation [23] each point along the filament so that e points along the
3
tangent of r(s) (see fig. 3). The generalized Frenet-Serret
(cid:18)L2(cid:19) 1 (cid:18)L (cid:19)2 equationstransportthematerialframealongthefilament,
π2 =F 0 + M2 0 , (1)
c A 4 c A
∂ e =Ω×e , (2)
s i i
where L is the rod length and A its bending rigidity [fig.
0
2(b)].Notethateq.(1)doesnotdependonthecompress- where∂ meansderivativewithrespecttos.Wecanchar-
s
ibility or the torsional rigidity of the rod. acterize any conformation by the angular strain vector
Similarbucklingorelasticinstabilitiesoccurinthedy- Ω = (Ω ,Ω ,Ω ) given in components with respect to
1 2 3
namicsofrodsatlowReynoldsnumber.Hereonetypically the local material frame. Along an ideal helical filament,
appliesatorqueatoneendofthefilament.Therodrotates spontaneous curvature κ and torsion τ are constant. As
0 0
Reinhard Vogel, Holger Stark: Motor-Driven Bacterial Flagella and Buckling Instabilities 3
thermaltorquem .Usingresistive-forcetheory,weintro-
th
ducelocalfrictioncoefficientsγ ,γ andγ (seeAppendix
(cid:107) ⊥ R
A) and arrive at the Langevin equations
(cid:2) (cid:3)
γ t⊗t+γ (1−t⊗t) v =f +f (6)
(cid:107) ⊥ el th
γ ω =m +m . (7)
R el th
Here v = ∂ r is the translational velocity, ω = ∂ φ the
t t
angular velocity about the local tangent vector t = e ,
3
and ⊗ means tensorial product. The anisotropic friction
tensor acting on v in Eq. (6) couples rotation about the
helical axis to translation and thereby creates the thrust
Fig. 3. The kinematic variables of a slender elastic rod are force that pushes the bacterium forward as illustrated in
the space curve r(s) of its center line and the material frame fig. 1 [28]. Experiments show reasonable agreement with
{e1,e2,e3} attached at each point of the center line. the approach of resistive-force theory [29,30]. Finally, the
thermal force f and torque m are Gaussian stochas-
th th
tic variables with zero mean, (cid:104)f (cid:105) = 0 and (cid:104)m (cid:105) = 0.
material frame for the equilibrium shape of the helical fil- th th
Their variances obey the fluctuation-dissipation theorem
ament, we choose the Frenet frame which consists of the
and therefore read
tangentvectort=e ,thenormalvectore =n=∂ t/κ ,
3 1 s 0
andthebinormalvectore2 =b=t×n.Thestrainvector (cid:104)fth(t,s)⊗fth(t(cid:48),s(cid:48))(cid:105)=2kBTδ(t−t(cid:48))δ(s−s(cid:48)) (8)
then becomes Ω = (0,κ0,τ0). Further parameters of an ×(cid:2)γ t⊗t+γ (1−t⊗t)(cid:3),
ideal helix are the pitch p = 2πτ /(κ2 +τ2) and radius (cid:107) ⊥
0 0 0
R = κ0/(κ20+τ02). The ratio of pitch and circumference, (cid:104)mth(t,s)mth(t(cid:48),s(cid:48))(cid:105)=2kBTδ(t−t(cid:48))δ(s−s(cid:48))γR, (9)
p/2πR=tanα, defines the pitch angle α. (cid:104)m (t,s)f (t(cid:48),s(cid:48))(cid:105)=0. (10)
th th
Thetotalelasticfreeenergyofthefilamentconsistsof
two contributions: In our simulations we use a discretized version of the
dynamic equations following our earlier work [31,22] (see
(cid:90) L
also Ref. [32,33,34]). We discretize the center line r(s) of
F = (f +f )ds (3)
cl st thefilamentbyintroducingN+1beadsatlocationsr(i) =
0
r(s=i·h)andwithnearest-neighbordistanceh.Toevery
The first term is Kirchhoff’s classical theory for bending bead we attach the material frame {e(i),e(i),e(i)} (i =
and twisting, 1 2 3
0,...,N) and approximate the tangent vector by
fcl = A2(Ω1)2+ A2(Ω2−κ0)2+ C2(Ω3−τ0)2, (4) e3(i) = |rr((ii))−−rr((ii−−11))|. (11)
where we introduced the bending rigidity A and the tor-
The transport of the material frame along the filament
sional rigidity C [23,27]. Instead of implementing a con-
occurs in two steps: First, we rotate about the bond di-
straint for the inextensibility of the filament in our sim-
rectione(i) byanangleΩ(i)htoimplementintrinsictwist
ulations, we also add a stretching free energy with line 3 3
plustorsion.Thereafter,weintroducethecurvatureofthe
density
filament, by rotating the the bond vector e(i) of the ma-
3
f = K (∂ r)2. (5) terial frame about e3(i) ×e(3i+1) by an angle (cid:112)Ω12+Ω22h
st 2 s into the consecutive direction e(i+1). With this procedure
3
thefreeenergydensitiesf andf fromEqs.(4)and(5)
We choose the spring constant K such that the changes cl st
are discretized and the functional derivatives of the total
in the filament length are below 1.5 %. The filament is
freeenergy,f =−δF/δr andm =−δF/δφ,reduceto
inextensible to a good approximation. el el
conventional derivatives with respect to r(i) and φ(i). In
addition,weapproximatethetangentvectorinthefriction
tensor in Eq. (6) by t(i) =(e(i)+e(i+1))/|e(i)+e(i+1)|.
2.2 Dynamics of the helical filament 3 3 3 3
We mostly performed deterministic simulations, only in a 2.3 The motor-driven hook
few cases we have added thermal fluctuations. We formu-
lateLangevinequationsforthelocationr(s)andintrinsic The flagellum is driven by a rotary motor embedded in
twistφ(s)ofthehelicalfilament.AtlowReynoldsnumber the cell wall of the bacterium. The motor torque is trans-
elastic force per unit length, f =−δF/δr, and thermal mitted to the helical filament by a short flexible coupling.
el
force f are balanced by viscous drag. The same applies Because of its shape it is called hook. With a well regu-
th
to the elastic torque per unit length, m =−δF/δφ and lated length of 0.05µm for E.Coli or S.typhimurium and
el
4 Reinhard Vogel, Holger Stark: Motor-Driven Bacterial Flagella and Buckling Instabilities
upto0.1µmforR.sphaeroides itismuchshorterthanthe
helical filament [35,36,37,38]. It is also shorter than the
discretization length of h = 0.2µm which we can employ
in our simulations as indicated in fig. 4. We, therefore,
cannot model the hook in full detail. Instead, we repre-
sent motor and hook by a motor torque that acts directly
on one end of the filament neglecting the extension of the
hook.
Moleculardynamicssimulationsshowedthatthehook
bends and twists easily. This is possible since conforma-
tional changes of molecular bonds require only a small
amount of energy [39]. So the hook itself allows the fil-
ament to nearly assume any orientation outside the cell.
Hence, it is comparable to a constant-velocity joint. The
blow-up in fig. 4 illustrates how motor and hook act to-
gether to drive the filament. The picture also shows the
rotationaldegreesoffreedomofthefilamentattheattach-
Fig.4.Blow-up:Thehookactsasauniversaljointbetweenthe
mentpointtothehook.Thefilamentcanrotateaboutits motor embedded in the cell wall and the long helical filament
localaxis,abouttheaxisparalleltothemotortorque,and which retains its full rotational degrees of freedom. Main pic-
towards or away from this axis. ture: The hook is much shorter than the discretization length
The task of the hook is to transmit the motor torque indicated by the blue and red segments of the filament. We
to the filament and to guarantee its rotational degrees of do not model the hook explicitly but let the motor torque act
freedom. In our coarse-grained model, we implement this directly on the first material frame of the filament which, in
taskbybalancingallthetorquesactingonthefirstmate- principle, can assume any orientation in space.
rialframe{e(0),e(0),e(0)}thatdeterminestheorientation
1 2 3
of one end of the filament:
(cid:2)γ he(0)e(0)+ 1γ h3(1−e(0)e(0))(cid:3)ω (12) 3 The motor-driven helical filament
R 3 3 2 ⊥ 3 3
=M −A[Ω e(0)+(Ω −κ )e(0)]−C(Ω −τ )e(0).
1 1 2 0 2 3 0 3
The material frame rotates with an angular frequency ω.
Itgivesrisetoafrictionaltorquedecomposedintoacom- Inthissectionwestudyindetailthethrustforcethatthe
ponentalongthetangentvectore(0) andperpendicularto motor-drivenhelicalfilamentgeneratesbothwhentheac-
3
it.Thelengthhappearsduetothediscretization.Thefric- tuated end of the filament is fixed in space or attached to
tional torque is balanced by the motor or external torque a larger load particle, which mimics the cell body. In par-
M = Me , which we assume constant throughout the ticular,wedescribethebucklingtransitionsbyillustrating
z
paper, and the elastic torque −δF/δΩ. the observed filament configurations.
Itisinstructivetoshortlylookatacompletelyrigidhe-
licalrodfirst,whichdoesnotexhibittranslationalmotion.
2.4 Simulation parameters
In the low Reynolds number regime, the angular velocity
ω of the rod and the applied torque M obey the linear
For the bending rigidity we use A = 3.5pNµm2 given relation M = Bω, where B is the rotational friction ten-
in Ref. [18] as a typical value for bacterial flagella and sor. For a long slender helix like the normal form of the
set it equal to the torsional rigidity, C = A. Our previ- bacterialfilament,oneprincipalaxisofBpointsalongthe
ous work showed that this is in agreement with experi- helical axis and the eigenvalues in the plane perpendicu-
mental observations [22]. All other parameters are deter- lartothisaxisaredegenerate,ingoodapproximation[40,
mined by the geometry. In our study we use the normal 41]. Now there is a formal analogy to the motion of the
stateofthebacterialflagellumwithspontaneouscurvature forceandtorquelessspinningtopwithaxialsymmetryin
κ1 = 1.3/µm and torsion τ1 = 2.1/µm. In the following classical mechanics [42,43]. We just replace the constant
westudyaright-handedhelicalfilamentalthoughthenor- torqueM bytheconservedangularmomentumandBby
mal state of a real flagellum is left-handed. We calculate the moment of inertia tensor. We explain details in ap-
thelocalfrictioncoefficientsfromLighthill’sformulas[40] pendix B. According to this analogy, the rigid helix in a
summarized in appendix A as γ(cid:107) = 1.6·10−3pNs/µm2, viscousfluidsprecessesabouttheconstantappliedtorque
γ =2.8·10−3pNs/µm2, and γ =1.26·10−6pNs, where while also rotating about its helical axis. However, in our
⊥ R
a filament diameter of about 20nm is used. The length of simulations we observe that as soon as we introduce a fi-
thefilamentisL=10µmcorrespondingtoapproximately niteelasticityofthehelicalrod,theprecessionisnolonger
four helical turns. The discretization length between the stableandthehelicalfilamentaligns,forexample,parallel
beads is chosen as h=0.2µm. to the torque.
Reinhard Vogel, Holger Stark: Motor-Driven Bacterial Flagella and Buckling Instabilities 5
(a) (i) (ii) (iii) (iv) (c)
(i)
e oscillating force
unstabl stable stable buckling cillatiknligng
s c
o u
b
(ii)
(iii)
0
(b)
(iv)
(iii)
(iv)
(ii)
(i)
Fig.5.(a)ThrustforceF versusmotortorqueM.Fourdifferentregimesassociatedwithdifferentconfigurationsoftherotating
filamentexist.Inregime(iii)and(iv)theminimumandmaximumvalueoftheoscillatingthrustforceareshown.Asupercritical
bifurcation occurs at the critical torque M ≈ 1.1pNµm indicating a buckling transition. A second bifurcation is visible at
c1
M ≈4.2pNµm.Theredlineinregimes(i)and(ii)followsfromresistiveforcetheory.(b)Thrustforceversustimeforspecific
c2
torquevaluesinthefourdifferentregimes:(i)M =−1.0pNµm,(ii)M =1.0pNµm,(iii)M =1.2pNµm,and(iv)M =4.5pNµm.
(c) Characteristic snapshots of the helical filament in the four regimes. The red circular arrow and ω indicate a rotation about
thelocalhelixaxisandFl thelocalthrustforce.Thegreencirculararrowandχshowtheprecessionabouttheexternaltorque
axis.Inaddition,thetrajectoryofthetipofthefirsttangentvectorisindicated:(i)Thegreenlinebelongstotheperpendicular
orientation of the filament, (iv) red line:fast rotation abouthelical axis, green line: slow precession about motor torqueduring
relaxation of the filament.
3.1 Force-torque relation and buckling types of configurations are available in the supplemental
material.
A negative torque M generates a negative force that
3.1.1 Discussion of the basic features
pulls at the anchoring point. However, we realized that
the orientation of the filament along the torque is not
The motor-driven helical filament creates a thrust force. stable. For long times the filament turns away from the
We calculate it as the force component on the first bead torque axis [green arrow in illustration (i) of fig. 5(c)] un-
paralleltotheappliedtorqueM =Me :F =−∂F/∂r(0)·et.ilitreachesaconfigurationperpendiculartoM,whereit
z z
We keep here the bead at a fixed position r0 = r(0) and slowly rotates about the local helical axis and slowly pre-
use the discretized version of the free energy (4). Figure cesses about M. This motion is also visible for the tip of
5(a) plots the resulting thrust force F versus the applied thefirsttangentvector.ThelinearincreaseofF withM in
torque determined in simulations without thermal noise. theregimes(i)and(ii)infig.5(a)fitswellwiththeresult
We discuss the graph in detail. fromresistiveforcetheoryforaperfecthelicalfilament,as
ApositivetorqueM producesathrustforcethatpushes indicated by the line (see appendix A). Small deviations
against the anchoring point of the filament. The thrust are visible at higher torques due to elastic deformations
force is constant in time as indicated by the straight line of the helix which enhance the thrust force.
(ii) in fig. 5(b). The illustration (ii) of fig. 5(c) shows the At a critical torque M ≈ 1.1pNµm the thrust force
c1
stable orientation of the helical filament along the torque starts to oscillate as curve (iii) in fig. 5(c) indicates. Min-
M. It rotates about the helical axis with angular fre- imum and maximum values of the force are plotted in fig.
quency ω. The local thrust force acting along the helix 5(a). They develop continuously from the constant force
axisisindicatedbyFl.Thetangentofthefilamentatthe at M indicating a supercritical Hopf bifurcation. Illus-
c1
anchoring point is tilted against M and the tip moves on tration(iii)offig.5(c)showsabuckledconfigurationthat
a circle, as indicated by the schematic. Movies for all four rotates about the local helix axis with frequency ω and
6 Reinhard Vogel, Holger Stark: Motor-Driven Bacterial Flagella and Buckling Instabilities
precesses with frequency χ about the motor torque M
(a) (b)
keeping its shape fixed. The trajectory of the tip of the
firsttangentvectorreflectsthismotion.Astraightforward
explanation is that the helical filament buckles under the
thrust force generated by the rotating filament. The force
adds up from the free to the fixed end of the filament
and puts the filament under compressional tension. This
issimilartoarodthatbucklesunderitsowngravitational
load [23,27]. In sect. 4 we will develop a theory for this
buckling transition which is quite involved. Finally, at a
critical torque value of M ≈ 4.2pNµm a second bifur-
c2
cation occurs in the force-torque relation of fig. 5(a). The
(c) (d)
buckled state itself becomes unstable, visible by the fast
oscillations of the thrust force in fig. 5(b). The buckled
configuration is compressed until the fixed end becomes
perpendicular to the motor torque. At this point the fast
rotation about the local helical axis stops and the thrust
force averaged over one fast period is approximately zero.
Now the strongly bent configuration of the filament re-
laxes slowly and precesses about the applied torque M
[second configuration in fig. 5(b)(iv)]. The thrust force on
theanchoringpointslowlyincreases.Whenthefilamentis
sufficientlyrelaxed,itstartsagainitsfastrotationsabout
the local helix axis and the whole cycle repeats. Fig. 6. (a) Relaxation rate λ of the elastic free energy ver-
sus applied torque M for a small disturbance of the aligned
statewherethefilamentisparalleltothetorquedirection.(b)
Angular velocity ω and precession frequency χ versus torque
3.1.2 Discussion of additional features
M.ThesupercriticalbifurcationatM isclearlyvisible.The
c1
redlineiscalculatedwithresistiveforcetheory.(c)Meanend-
The reported supercritical Hopf bifurcation is also visible
to-end distance (cid:104)r(cid:105) in units of the helix length L versus M.
0
in other quantities besides the thrust force. We discuss
The red dots are results from Brownian dynamics simulations
here additional properties of the motor-driven helical fil-
withthermalnoiseincluded.(d)Standarddeviationσ(r)ofthe
√
ament.
end-to-enddistanceinunitsofσ =R/ 2versusM.Thermal
0
To quantify the stability of the filament aligned par- noise (red dots) leads to fluctuations about the mean value.
allel to the motor torque axis, we recorded the temporal
evolution of the elastic free energy starting from a small
disturbance of the aligned state and fit it to the form for rotations of the whole filament about the torque axis
is plotted in the inset. A non-zero χ corresponds to the
|F −F0|≈δF0exp(λt)sin(ωt). (13) buckled state.
Finally, figs. 6(c) and (d) plot the mean end-to-end
Hereω istheangularvelocityoftherotatinghelixleading distance (cid:104)r(cid:105) of the helix and its standard deviation σ as a
to oscillations in F and λ is the reorientation rate. The function of M, respectively. They are defined as
resultforλisplottedinfig.6(a).ForpositiveM belowthe
critical torque, the negative λ indicates the stable aligned 1 (cid:90) T
(cid:104)r(cid:105)= lim |r(s=L)−r(s=0)|t. (14)
state.ForsmallM areorientationofthefilamentcouldnot T→∞T 0
be detected within the simulation time. Frictional forces σ2 =(cid:10)(r−(cid:104)r(cid:105))2(cid:11). (15)
are small and hardly deform the helix which, therefore,
just precesses about the applied torque. Nevertheless, to Whereas (cid:104)r(cid:105) is continuous at both bifurcations, the stan-
record the thrust force-torque relation, we always started darddeviationdisplaysapronounceddiscontinuityatthe
from an aligned state at M = 1pNµm and then changed second bifurcation in agreement with the beha√vior of the
the driving torque to the desired value and let the elastic thrust force. We write σ in units of σ = R/ 2, where
0
free energy relax to its stationary value, where we finally R is the helix radius. σ is the maximum value of σ in
0
recorded the thrust force. The small positive λ for M <0 regime (iii) where the buckled helix has a constant shape
indicatestheslowreorientationofthefilamenttowardsthe but the free end of the filament rotates on a circle with
perpendicular configuration The supercritical Hopf bifur- radiusR.Thestrongincreaseofσ inregime(iv)isdueto
cation is located where λ changes sign from negative to the oscillating buckled state.
positive. The rotating filament also experiences thermal forces
Figure 6(b) shows the angular frequency ω for rota- duetotheviscousenvironment.However,sincethepersis-
tions about the local helix axis as a function of M. The tence length A/k T ≈1mm calculated from the bending
B
linearregimebelongstothealignedstate,deviationsfrom rigidityAismuchlargerthanthefilamentlengthof10µm,
it occur in the buckled state. The precession frequency χ we do expect that our results are robust against thermal
Reinhard Vogel, Holger Stark: Motor-Driven Bacterial Flagella and Buckling Instabilities 7
fluctuations.Thisisconfirmedbytheend-to-enddistance
(a)
(cid:104)r(cid:105) in fig. 6(c) (red dots) which agrees with the determin-
isticsimulations.Thestandarddeviationin6(d)indicates
some fluctuations. Below the buckling transition we can
directly connect them to compressional fluctuations us-
ing the spring constant of the helical filament, A/(R2L),
calculatedinourearlierarticle[22].Theequipartitionthe-
orem gives σ/σ ≈ 0.15 in good agreement with the sim-
0
ulated value of 0.18. In the buckled state, the helical fil-
ament has more opportunities to fluctuate around which
explainsthefurtherincreaseofσ.Furthermore,weobserve
strong fluctuations of the thrust force in our simulations
(b)
which result from the delta correlated stochastic forces unstable un
acting on the fixed first bead of the discretized filament. s
t
An average over these fluctuations agrees with the deter- ab
l
ministic case (data not shown). The fluctuations will also e
be smoothed out in an experiment which performs some
temporal average during measurement. stable
3.2 Buckling instability during locomotion
So far we have studied the situation where one end of the
filamentisfixedinspacesothatitcannottranslate.How-
Fig. 7. Buckling transition for a helical filament attached to
ever, rotating flagella push the cell body of a bacterium
a bead of radius a that can move along the z direction. (a)
forward so that it moves. We mimic this scenario by at-
Critical torque M as a function of inverse bead radius 1/a.
taching the filament to a bead of radius a which, for sim- c1
Inset:Blow-upforthebiologicallyrelevantregime.(b)Critical
plicity, can only move along the z direction. The thrust
force F versus critical torque M .
force F generated at the attached end of the filament c1 c1
is then used to push the sphere forward acting against
the Stokes friction force. We observe similar thrust force- axis,respectively.Thisformulawiththecoefficientscalcu-
torque relations as for the case of a fixed filament. The latedbyresistiveforcetheory(seeappendixA)reproduces
aligned state is again unstable for negative torque and the linear increase for small 1/a, as demonstrated by the
possesses a larger reorientation rate which might have bi- red line in the inset of fig. 7(a).
ological relevance as we discuss in sect. 5. For positive In fig. 7(b) we plot the critical thrust force versus the
torque, the aligned state is stable and the thrust force critical torque. For biologically relevant values M be-
c1
grows linearly in the driving torque M until the Hopf tween 1 and 2pNµm the critical force is indeed nearly
bifurcation occurs at a critical value Mc1 indicating the constant. It only shows a very slow linear increase since
buckling instability. frictional forces due to the motion of the helix stabilize it
Figure 7(a) shows the critical torque Mc1 as a func- against buckling. At Mc1 ≈4pNµm the behavior changes
tion of the inverse bead radius 1/a. From 1/a=0, which dramatically.Thecriticalthrustforcegoestozeropropor-
corresponds to the fixed filament, the critical torque in- tional to M2 (see dotted line) following the behavior of a
creases linearly in 1/a and then at a−1 ≈ 5/2 turns into rod that bucc1kles under an applied force and torque as de-
a slow growth towards the value for the freely swimming scribedintheintroduction.Inthisregimethesupercritical
helix, i.e., 1/a → ∞. In the biological relevant case with Hopf bifurcation becomes subcritical and hysteresis oc-
the cell body size a ≈ 0.5···2µm, the linear dependence curs.Sowhereasforsmalltorquesbucklingishinderedby
of the critical torque on 1/a can be derived based on the locomotion, for large torque the typical quadratic depen-
fact that the critical thrust force Fc1 is nearly constant, dence Fc1 ∝Mc21 is observed. In the following section, we
as we show in fig. 7(b). So the velocity v =Fc1/(6πηa) is develop a theory to describe the observed buckling tran-
so slow that the buckling transition is hardly influenced sition.
by the motion of the helical filament with the attached
bead. Now, force and torque on the helix depend linearly
on velocity v and angular velocity ω (see appendix A). 4 Buckling theory for a helical rod
Eliminatingω andsettingv =F /(6πηa)atthebuckling
c1
transition, one arrives at
The goal in this section is to formulate a theory that re-
B (cid:18) A B (cid:19) F 1 produces the force-torque relation in fig. 7(b) for the first
Mc1 =−C(cid:107)Fc1+ C(cid:107)− C(cid:107) (cid:107) 6πc1ηa. (16) buckling transition of the helical filament as obtained by
(cid:107) (cid:107) our simulations. Clearly, this relation cannot directly be
Here A , B , and C are the translational, the rotational, explained by the theory of a thin elastic rod that buckles
(cid:107) (cid:107) (cid:107)
and coupling friction coefficients parallel to the helical under the influence of an external force and torque which
8 Reinhard Vogel, Holger Stark: Motor-Driven Bacterial Flagella and Buckling Instabilities
and torque densities due to the applied motor torque and
friction with the surrounding fluid. In addition, bound-
ary conditions are necessary. At the free end of the rod
(s=L ) no external force and torque act, so elastic force
0
andtorquehavetovanish.Theendattachedtothesphere
Fig. 8. The helical filament is approximated by a thin helical can only move in z direction with velocity v. The exter-
rodoflengthL0 thatischaracterizedbytheeffectivebending nal torque Me is balanced by the elastic torque M(0)
z
rigidity A and local friction coefficients of the helical fila-
eff and the thrust force on the sphere F =6πηav equals the
ment. The applied torque Me generates the frictional torque
z elastic force at the leading end of the rod, F (0):
z
m and forces f and f .
(cid:107) ⊥
F (0)=F =6πηave , M(0)=Me , (19a)
z z z
F(L )=0, M(L )=0. (19b)
weshortlymentionedintheintroductioninEq.(1).There 0 0
are several reasons for this. First, the helical filament is
not just a simple elastic rod. Second, the external force
After setting up the problem, we have to explain how
that puts the helix under tension is generated locally by
the different forces and torques entering Eqs. (18) look
the rotation-translation coupling of the helix and accu-
like for the helical rod close to the buckling transition.
mulated along the filament similar to a rod that buckles
TheelastictorqueM isproportionaltotheangularstrain
under its own gravitational weight. Third, the whole fil-
vector Ω written in components with respect to the local
ament moves with a constant velocity which leads to ad-
material frame {e ,e ,e }:
1 2 3
ditional frictional forces and it also precesses about the
external torque in the buckled state. In the following we M =A Ω e +A Ω e +C Ω e , (20)
eff 1 1 eff 2 2 eff 3 3
formulate a model based on the theory of a thin elastic
where A is the effective bending rigidity of Eq. (17).
helical rod, derive from it a force-torque relation for the eff
Sincebucklingtheoryconsiderslocaldisplacementsofthe
buckling transition, and compare it to fig. 7(b).
rod only, the torsional term and the actual value of the
effectivetorsionalrigidityC arenotimportant.Thefor-
eff
4.1 Model equations mulation for M is in full analogy to our presentation in
sect. 2.1, only the spontaneous curvature and torsion are
Tosetupourmodelequations,weapproximatethehelical zero for the helical rod which serves as an effective repre-
filament by a thin helical rod where the helicity comes in sentation of the helical filament. In setting up linearized
through the rotation-translation coupling in the friction equations in the vicinity of the buckling transition, the
matrix. The length of the rod, L = sinαL, agrees with elastic force F is only needed for the unbuckled straight
0
the height of the helix, where α is the pitch angle. In rod oriented along ez. Since the external force density f
engineeringsciencethebucklingofhelicalspringsisawell is constant for the straight rod, as we argue in the next
known problem. If the height of the spring is larger than paragraph, Eq. (18a) and boundary conditions (19) give
its radius, one approximates the spring by a soft rod with the linear force profile
effective bending, shear, and compressional rigidity [44,
F(z)=f (L −z)e with f =F/L , (21)
45,46]. In our case, in contrast to classical helical spring (cid:107) 0 z (cid:107) 0
theory,thepitchofthehelixismuchlargerthanitsradius. whereweintroducef asthrustforceF dividedbytherod
(cid:107)
Wethereforehadtogeneralizethetheoryofhelicalsprings lengthL .Wewilluseitasoneparameterinthefollowing.
0
in Ref. [46] to derive an effective bending rigidity of the The straight filament moves with a constant veloc-
helical rod in terms of the bending and torsional rigidity ity ve and rotates with a constant angular velocity ωe .
z z
of the filament: Theyresult,respectively,inaconstantfrictionalforceden-
(cid:18) (cid:19) sity f e and a torque density me with
1 1 1 1 A (cid:107) z z
= 1+sin2α+ cos2α (17)
A 2Asinα C
eff f =a v+c ω, (22a)
(cid:107) (cid:107) (cid:107)
Details of the derivation are given in appendix C. m=c(cid:107)v+b(cid:107)ω, (22b)
To address buckling of the helical rod, we start with
where the frictional coefficient c couples translation to
the balance equations for force and momentum acting on (cid:107)
rotation. Appendix A gives the coefficients a , b , and c
a thin elastic rod [23,27] and neglect any inertial contri- (cid:107) (cid:107) (cid:107)
forthehelicalrodintermsoftheparametersofthehelical
bution in the low Reynolds number regime:
filament. In Eq. (21) we have already linked f to the
(cid:107)
F(cid:48)+f =0, (18a) thrust force F. From Eq. (18b) and boundary conditions
(19), we also deduce a linear torque profile
M(cid:48)+e ×F +m=0, (18b)
3
M(z)=m(L −z)e with m=M/L , (23)
where (cid:48) means derivative with respect to the arc length 0 z 0
s and e is the local tangent vector. Here F and M wherewerelatemtotheappliedmotortorqueM divided
3
are internal elastic forces and torques acting along the by the rod length L . So, m is the second parameter in
0
rod,whereasf andmdenote,respectively,externalforce our problem.
Reinhard Vogel, Holger Stark: Motor-Driven Bacterial Flagella and Buckling Instabilities 9
The buckled rod after the first buckling transition in Here we introduced the rescaled coordinate zˆ = z/L
0
oursimulationshasaconstantshape.Itrotatesaboutthe and the dimensionless parameters mˆ = mL2/A , fˆ =
0 eff (cid:107)
local tangent vector with angular velocity ωe3 and pre- f L3/A , fˆ =f L3/A , and χˆ=χa L4/A .
cesseswithangularvelocityχabouttheaxisoftheapplied (cid:107) 0 eff ⊥ ⊥ 0 eff ⊥ 0 eff
Equations (29) are quite general and several related
torqueleadingtoalocalvelocityχe ×r.Furthermore,the
z problems follow from them. When forces fˆ and fˆ van-
filament translates with velocity v along the z-direction (cid:107) ⊥
ish, they describe the writhing instability of rotating rods
andthetotallocalvelocityamountstov =ve +χe ×r.
z z [24]. For zero torque and precession frequency, and fˆ =
Inthevicinityofthebucklingtransition,deformationsare (cid:107)
small and in leading order we can identify v and ω with −fˆ = fˆ, one arrives at the classical example of a col-
⊥ z
the values of the straight rod. Then, the frictional torque umn that buckles under its own weight [23,27]. A similar
along the local tangent vector is problem occurs for microtubuli that buckle under the ac-
tion of molecular motors [47]. In our case, the force den-
m=me , (24)
3 sity f that causes buckling points along the rod axis and
(cid:107)
f stabilizes the straight rod for non-zero v. In compari-
where m is already given in Eq. (22b). So, close to the ⊥
son, the column under gravity always experiences a force
buckling instability we can identify m with the applied
density along the vertical which gives a force component
motor torque as in Eq. (23). The frictional force density
perpendiculartotherodassoonasitbucklesandthereby
becomes
supports buckling.
f =f P e +f P e +a χP (e ×r), (25) Wecompletethelinearizeddynamicequations(29)by
(cid:107) (cid:107) z ⊥ ⊥ z ⊥ ⊥ z
writingtheboundaryconditions(19)inlinearizedandre-
where we use the projectors
duced form:
P(cid:107) =e3⊗e3 and P⊥ =1−e3⊗e3 (26) X(0)=0 Y(0)=0, (30a)
onthedirectionsparallelandperpendiculartothetangent X(cid:48)(cid:48)(0)=−mˆY(cid:48)(0) Y(cid:48)(cid:48)(0)=mˆX(cid:48)(0), (30b)
vector e3. The force density f(cid:107) has already been given in X(cid:48)(cid:48)(1)=0 Y(cid:48)(cid:48)(1)=0, (30c)
Eq. (22a) and
X(cid:48)(cid:48)(cid:48)(1)=0 Y(cid:48)(cid:48)(cid:48)(1)=0. (30d)
f =a v (27)
⊥ ⊥
The first line means that the attached end of the rod can
characterizesthefrictionalforcedensitygeneratedperpen-
only move in z direction and not along the x and y axis.
dicular to the local rod axis when the rod moves with ve-
The second line means that a torque does not act per-
locityv.Sincethefrictionalcoefficienta islargerthana ,
⊥ (cid:107)
pendicular to the z-axis. So if the rod starts to buckle,
f actsagainstbuckling.Finally,a χisthefrictiondueto
⊥ ⊥
the local torque me has to be equilibrated by a bending
theprecessionoftherod.WenotethatatermP ·(e ×r) 3
(cid:107) z
moment.Thefreeendoftherodistorquelessand,there-
doesnotappearsinceitdoesnotcontributeinleadingor-
fore,theroddoesnotbend,asexpressedbythethirdline.
dertof .Wealsodidnotincludetherotation-translation
(cid:107)
Finally, the free end is also force free and the fourth line
coupling perpendicular to e since the two terms cancel
3
follows from Eq. (18b) by setting F =0.
eachotherintheequations,weformulateinthefollowing.
To search for nontrivial solutions of Eqs. (29) in our
We will analyze the buckling transition by first con-
parameter space and thereby identify the buckling transi-
sidering the four parameters m,f ,f , and χ as indepen-
(cid:107) ⊥
tion, we proceeded as follows. In addition, to the bound-
dent and then apply our results to reproduce the force-
aryconditions(30a)and(30b),nontrivialsolutionsofthe
torque relation of the helical rod. Buckling occurs when
buckling equations (29) can be characterized by X(cid:48)(0),
the straight solution r(z)=(0,0,z) of Eqs. (18) becomes
Y(cid:48)(0), X(cid:48)(cid:48)(cid:48)(0) and Y(cid:48)(cid:48)(cid:48)(0). The principal idea is to use
unstable and a new non-trivial solution occurs at a cer-
them to generate solutions of Eqs. (29) and to fulfill the
tain parameter set. We, therefore, use the ansatz r(z) =
boundary conditions (30c) and (30d) at the free end by
(X(z),Y(z),z) and seek two equations linear in X, Y,
varying them. However, since X(cid:48)(0) and Y(cid:48)(0) just deter-
and its derivatives. We start by taking the derivative of
Eq. (18b) and use F(cid:48) =−f to arrive at minetheamplitudeofabentconfigurationandmerelyfix
therotationaldegreeoffreedomaboutthezaxis,theycan
M(cid:48)(cid:48)+e(cid:48) ×F −e ×f +m=0, (28) be chosen arbitrary. Instead, we vary two of our four pa-
3 3
rameters,fˆ andχˆ,tofulfillthefourboundaryconditions
where we insert the concrete formulas for M, F, f, m. ⊥
atzˆ=1.Asaresult,forgivenmˆ andfˆ,wedeterminepa-
We linearize these resulting equations using the identi- (cid:107)
ties Ω1 ≈ −Y(cid:48)(cid:48), Ω2 ≈ X(cid:48)(cid:48), P(cid:107)·ez = e3 ≈ (X(cid:48),Y(cid:48),1), rameters fˆ⊥ and χˆ for which non-trivial solutions of the
P ·e ≈ −(X(cid:48),Y(cid:48),0), and P ·(e × r) ≈ (−Y,X,0), buckling equations exist and thereby identify the mani-
⊥ z ⊥ z
and ultimately arrive at foldofbifurcationpointsinourfour-parameterspace.We
discuss it in the following section.
0=−Y(cid:48)(cid:48)(cid:48)(cid:48)+∂ (mˆ(1−zˆ)X(cid:48)(cid:48))
z
−fˆ(1−zˆ)Y(cid:48)(cid:48)−fˆ Y(cid:48)+χˆX, (29a)
(cid:107) ⊥ 4.2 Discussion
0=X(cid:48)(cid:48)(cid:48)(cid:48)+∂ (mˆ(1−zˆ)Y(cid:48)(cid:48))
z
Figure 9(a) plots the manifold of bifurcation points. To
+fˆ(cid:107)(1−zˆ)X(cid:48)(cid:48)+fˆ⊥X(cid:48)+χˆY. (29b) each parameter triple (mˆ,fˆ,fˆ ) belongs a specific value
(cid:107) ⊥
10 Reinhard Vogel, Holger Stark: Motor-Driven Bacterial Flagella and Buckling Instabilities
(a)
(c)
b
uc
kl
e
d
straight
d
e
straight uckl
b
tilted
(b) (d)
simulation
straight undistorted helix
distorted helix
Fig. 9. (a)Manifoldofbifurcationpointsintheparameterspace(mˆ,fˆ,fˆ ).Toeachparametertriplebelongsaspecificvalue
(cid:107) ⊥
of the precession frequency χˆ. (b) Buckling curves f (m) for different values of the perpendicular force ranging from fˆ =0 in
(cid:107) ⊥
steps of 25 to 300. The red dots are the critical forces and torques from fig. 7(b). (c) The plane pictures relation (31) between
f , f , and m for the helical rod with constant friction coefficients. Intersecting it with the manifold of bifurcation points
(cid:107) ⊥
gives the buckling curves in (d). Full blue line: for constant friction coefficients of the undistorted helix, dashed green line:
torque-dependent friction coefficients.
of the precession frequency χˆ which we do not discuss in steps of 25 to 300. At fˆ = 0 the typical parabolic
⊥
further here. At positive fˆ and for small mˆ and fˆ the curve of Eq. (1) occurs. At constant but small value of
⊥ (cid:107)
straight configuration of the helical rod is stable. If we m, the critical force f(cid:107) increases strongly with increasing
changethesignoffˆ ,abifurcationoccurswhichweinter- fˆ . Likewise, one needs large forces fˆ to stabilize the
⊥ ⊥ ⊥
pret as an instability of the straight rod when it reorients straight helical rod at high torques. The red dots are the
towards the perpendicular configuration. We saw this in- criticalforcesandtorquesfromfig.7(b)determinedinour
stability in our simulations when we reversed the driving simulations. We plot them in reduced units where we cal-
torque as discussed in sects. 3.1.1 and 3.2. Here we keep culate A from Eq. (17). Note that the buckling curves
eff
the direction of the torque but reverse the sign of the ve- develop a shoulder at mˆ around 15 for increasing fˆ due
⊥
locityvandtherebythesignoffˆ inEq.(27)byreversing to the ridge in the manifold of bifurcation points in fig.
⊥
the chirality of the rod. The main result is the surface in 9(a). The two simulation points at large m are close to
dark yellow that belongs to the first buckling transition this ridge. We speculate that the transition from a super-
observedinoursimulations,soatlargemˆ therodisbuck- critical bifurcation observed in our simulations at low m
led. Finally, at fˆ ≈ 0 and large mˆ a transition between to a subcritical bifurcation at large m is connected to the
⊥
twodifferentconfigurationsofthebuckledrodoccurs.An existence of this ridge.
interesting feature is the ridge in the bifurcation surface.
However, we could not determine any dramatic changes
in the buckling of the helical rod close to this ridge.
Intherotatinghelicalfilamentorhelicalrod,theforces
Figure 9(b) shows buckling curves fˆ(cid:107)(mˆ) for different f(cid:107) and f⊥ and the torque m are related to each other by
values of the perpendicular force ranging from fˆ = 0 Eqs. (22) and (27). Eliminating velocity v and angular
⊥