WHY ARE FLAGELLA HELICAL?
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1
S. Clark , *R. Prabhakar
1
Department of Mechanical & Aerospace Engineering
Monash University, Clayton, VIC-3800, Australia
* [email protected]
ABSTRACT
Flagella are novel devices that facilitate motility in micro-scale organisms. These semiflexible filaments are often intrinsically helical. Recent studies have revealed that
propulsion in a viscous environment is possible even with simpler achiral filaments
which, when twirled at one end, undergo a wrapping transition to a twisted
configuration. Simulations have shown that introducing a small amount of intrinsic
helicity eliminates the sharp transition and associated hysteresis. Here we explain why
this is the case, using a simple two link model. We also show that intrinsic helicity
assists in giving smoother thrust characteristics, both for isolated filaments and bundled
pairs.
INTRODUCTION
Life at the micro-scale is very different to that at the macro-scale. Viscous forces
dominate over inertia, and propulsion mechanisms employed by larger creatures are
ineffective for microorganisms. For example, reciprocating motion such as the opening
and closing of a scallop or flapping of a stiff propeller is unable to produce net
displacement in this regime (Purcell, 1977).
Microorganisms have therefore developed novel ways of achieving motility. Many
bacteria employ semi-flexible helical filaments called “flagella”, driven by motors at
their base. Some are monotrichous, such as Vibrio alginolyticus, possessing a single
polar flagellum, which drives the cell like a corkscrew. Others, such as Escherichia coli,
are peritrichous, and possess several flagella located all over the cell body. When rotated
in synchrony, hydrodynamic interactions cause these flagella to bundle together to form
a single helix, resulting in smooth forwards propulsion. Understanding these different
techniques is a central aim in the field of micro-scale motility.
Complementary to understanding biological flagella is the problem of building artificial
micro-scale swimming robots, and design of appropriate actuation mechanisms. The
physical principles behind these methods are therefore attracting widespread interest
among physicists, biologists and engineers alike.
In this work we try to understand the effect of shape – in particular helicity – on the
effectiveness of tethered flexible rotating filaments for use in propulsion. This stems
from the observation (Manghi et al 2006, Qian et al, 2003) that a straight flexible
filament making a fixed angle to the vertical at its base is capable of enabling motility
when driven at the base by an externally imposed torque. At small torques, the filament
deforms slightly, rotating on the surface of an imaginary cone and producing negligible
S. Clark, R. Prabhakar
thrust. Above a critical torque, however, the filament’s free end begins to deflect
towards the central axis, with the filament taking on a more curved conformation.
Associated with this transition is a jump in the angular velocity and thrust, and a strong
hysteresis. Bacterial flagella on the other hand, being naturally helical, are not observed
to undergo any such transition.
In recent work (Clark & Prabhakar, 2011), we began to bridge the gap between
dynamics of straight and helical flagella in order to understand what advantages the
latter may have. We showed through simulations that even a small amount of intrinsic
twist in filaments eliminates the sharp transitions and suggested that this could lead to
smoother operating characteristics in twisted filaments (Fig. 2 – symbols).
Here we explain why this is the case by comparing the results of simulations with a
simple two-link model of an intrinsically twisted filamentous propeller. The following
section outlines the simulations and we subsequently analyse the behaviour with the
lumped-parameter model. We further consider how intrinsic helicity affects thrust
forces in single filaments, as well as in a pair of filaments undergoing bundling.
STOKESIAN DYNAMICS SIMULATIONS
In simulations, the rest-shapes of our filaments are right-handed helices (Fig. 1). The
helical radius R and pitch P are related to the filament curvature κ and complimentary
helix angle ψ made by the tangent at any point on the contour to the z-axis, by
κ = R (R 2 + P 2 ) and ψ = tan −1 (R P ), respectively. The angle ψ is thus also the angle the
filament makes at its base to the vertical. We compare how κ changes the behaviour of
filaments whose contour length L, angle ψ and hydrodynamic thickness 2a are all held
fixed. Thus, while helices with larger values of κ have smaller radii and pitch, all of
them have the same vertical height Z max = L cos(ψ ) at rest.
Fig. 1: a) (i) Top-view of rest shapes of intrinsically twisted filaments as a family of
helical space curves all of same contour length that lie on surfaces of cylinders of
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S. Clark, R. Prabhakar
different radii. The cylinder surface is shown as a grey circle for one of the helices (pink
curve). (ii) Isometric view of the pink curve in (i) along with its cylinder. b) Schematic
of the discretisation of right-handed helix into segments. The arrows indicate connector
vectors. The angle ξµ is that subtended by the two planes intersecting along the
connector qµ, while χµ is the angle between vectors qµ and qµ+1.
Each filament is then discretised as a chain of N b = L / (2a) point particles, each with
a bare friction coefficient of ζ = 6π aη , where η is the shear viscosity of the
surrounding Newtonian fluid. As in our previous study, Brownian fluctuations and the
presence of external boundaries are neglected. In the zero-Reynolds-number limit, the
position of the µ - th “bead” on the chain is governed by an ordinary differential
equation, drµ dt = ∑ N M µν ⋅ Fν , where µ = 1 the tensor M µν =ζ −1[δ µν I+ζ ž (rµ − rν )] .
b
Here, I is the unit tensor, δµν is the Kronecker delta function, ž (x ) is a tensorial
function that describes the hydrodynamic interactions between a pair of beads separated
by the vector x. We use here the Rotne-Prager-Yamakawa regularisation of the freespace Oseen-Burgers interaction (Rotne et al, 1969, Yamakawa, 1970). The total nonhydrodynamic force on the µ-th bead, Fµ, is the resultant of elastic, excluded-volume
and external forces on that bead. The elastic force FµU = − dU drµ is calculated from the
intra-chain potential energy,
Nb − 2
2
2
1 Nb −1
1 Nb − 3
U = U 0 + ε s ∑ qµ − 2a −ε b ∑ cos χ µ − χ 0 + ε t ∑ ξµ − ξ0 ,
2 µ =1
2 µ =1
µ =1
(
)
(
)
(
)
(1)
accounting for the resistance of the filament to change from its rest shape. Here, qµ is
the vector connecting the µ-th bead to the µ+1-th bead, χ µ is the angle between qµ and
qµ+1, and ξµ is the angle between the pair of successive planes defined by qµ and qµ+1,
and qµ+1 and qµ+2 (Fig. 1b). Denoting unit vectors with a “ ˆ ”, these angles are
determined as χ µ = cos−1 (q̂ µ ⋅ q̂ µ +1 ), and ξµ = σ cos−1 (V̂ ⋅ Ŵ ) (Bekker, 1995), where
(
)
σ = sgn q µ ⋅ (q µ +1 × q µ +2 ) .
V = −q µ ⋅ (I − q̂ µ +1q̂ µ +1 ) is a vector normal to qµ in the plane spanned by qµ and qµ+1 and
similarly W = q µ +2 ⋅ (I − q̂ µ +1q̂ µ +1 ) is in the plane of qµ+1 and qµ+2 (Fig. 1). The constant
angles χ0 and ξ0 specify the zero-energy rest shape of the helix, and are calculated by
noting that the change in polar angle from one bead to the next is approximately
2κ a cosec (ψ ) . The constants ε s , ε b and ε t in the elastic energy function are linked to
the stretching, bending and twisting stiffnesses respectively and are related by
ε s a 2 4 = ε b = 2ε t . Excluded-volume forces are implemented using a pair-wise
repulsive potential between beads U EV = ε b (a / x ) acting when the distance between
any pair x < 2a . These interactions are important for pairs of filaments that can
approach each other closely during bundling.
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S. Clark, R. Prabhakar
The basal end of the filament in our simulations is located at (x,y,z)=(0,0,0). The first
bead µ =1 is connected to it such that it maintains the filament shape, and the angle its
position vector makes to the vertical (z-axis) is ψ. To this end, we use a constraining
potential U c = −ε b cos  cos−1 (r̂1 ⋅ z ) − ψ  + 12 ε s (r1 − 2a ) − ε t cos  cos−1 (r̂1 ⋅ q̂1 )− χ 0  + ε t (ξc − ξ0 ) ,
2
1
2
2
where ξc = σ cos−1 (V̂c ⋅ Ŵc ), Vc = −r1 ⋅ (I − q̂1q̂1 ) and Wc = q2 ⋅ (I − q̂1q̂1 ) . Filaments are twirled
in the clockwise direction (looking down from the top towards the base) by an external
torque T = -Tẑ, T > 0 , imposed as an external point force at the instantaneous position
of the first bead.
All quantities are rescaled by the space, time and energy scales, L , ζ L2 / ε b , and ε b ,
respectively. For all the results reported here, Nb = 20, and ψ = π / 4 (i.e. R = P). The
input parameters into our simulations are the dimensionless curvature κ, and the
magnitude of the torque T. The set of 3N b differential equations for bead positions are
integrated using a predictor-corrector scheme. For sufficient numerical accuracy we
choose time steps in the range ∆t = 10 −3 − 10 −5 . Simulations are run until the angular
velocity of chain centre-of-mass ω attains a steady value. For any choice of κ, the initial
states are either the rest-shapes, or steady-state shapes obtained at a very high imposed
torque.
WHY DOES THE HYSTERESIS AND TRANSITION DISAPPEAR?
Figure 2a below shows previous results showing the angular velocity as a function of
the applied torque. This demonstrates the existence of the transition reported previously
in straight filaments (triangles) (Manghi et al, 2006, Qian et al, 2003): at low imposed
torques, the filament rotates very slowly, whereas at high torques, it rotates much more
quickly.
Fig. 2: a) (Reproduced from Clark and Prabhakar, 2011) Steady-state rotational velocity
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S. Clark, R. Prabhakar
as a function of applied torque in single filaments of different intrinsic curvatures.
Filaments of lower intrinsic curvature ( , κ = 0; , κ = 1) exhibit two different steadystates over a window of applied torques, whereas no bistable behaviour is observed at
larger curvature (●, κ = 1.42). Lines are drawn to guide the eye. b) Steady state shapes
of straight and curved filaments at two distinct torques.
There exists a range of intermediate torques for which an intrinsically straight filament
is bistable, and depending on the initial conditions, takes on either a slowly rotating
nearly straight shape or a more quickly rotating “wrapped” configuration (see Fig. 2b).
The boundaries of the bistable region are marked by sharp jumps in angular velocity. As
the inherent curvature κ in the filament is increased, the size of the bistable domain
shrinks (squares) and then disappears entirely (circles), although a jump in the angular
velocity persists.
Qian et al (2003) introduced a lumped-parameter model to explain the wrapping
transition in the case of a straight flexible filament, and we extend this model to
understand the effect of intrinsic curvature.
Fig. 3: a) Projections of the native configuration of the two-link model on three
orthogonal planes. The solid circles represent the location of the drag forces on the
filament due to the rotational flow of the surrounding fluid. b) Angular velocity as a
function of applied torque (blue, φ/θ = 0, magenta, φ/θ = 0.2, yellow φ/θ = 1). As this
intrinsic curvature increases, the hysteretic transition disappears.
The filament is modeled as two straight links (Fig. 3 a), each of fixed unit length, and
connected by a torsional spring with stiffness K. The first link OP is set at an angle of θ
to the z-axis. The native state of the second link PQ0 defines the intrinsic helicity. For a
straight filament, this would be parallel to OP (as in Qian, 2003). For non-zero
curvature, we first rotate this vector at P by a small angle –φ in the y-axis, and then
rotate the projection in the x-axis by the same (small) angle (see Fig. 3a). Therefore, in
general for small angles the native rest shape of the filament is given by
PQ 0 ≈ (θ − ϕ , ϕ ,1) .
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S. Clark, R. Prabhakar
A rotational flow is then imposed about the z-axis, with angular velocity ω, and resultant
velocity field v (r ) = ω ẑ × r at position r. (We do not require pseudo-forces to correct
for being in the filament’s rotational frame of reference, as inertia of both the fluid and
the filament is negligible.) Under the imposed flow, the link OP is held fixed, but PQ is
free to pivot about P from its position at rest and the steady-state coordinate of Q is
given by OQ = (x, y, 2) . The steady-state configuration is determined by a balance of
moments of the frictional drag forces and the resistance of the torsional spring. The
moment on PQ about the torsional spring is Ts ≈ K(y − ϕ , 2θ − x − ϕ , 0) . Assuming all
of the drag on PQ is located at Q with associated drag coefficient ζ, the moment on PQ
due to hydrodynamic drag is given by Tf = −PQ × ζ v Q ≈ ζω (x, y, 0 ).
Balancing these moments gives K(y − ϕ ) = ζω x and K(2θ − x − ϕ ) = ζω y , which can
be solved for the unknowns x and y hence x = (2θ − ϕ ) − (ζω / K )ϕ and
1 + (ζω / K ) y = (2θ − ϕ )(ζω / K ) + ϕ . Finally, assuming that the drag on OP is located
at P’ ≈ (dθ, 0, d) the total drag on the whole model is given by
T = OP '× ζ v P ' + OQ × ζ v Q , so that the net rotational drag about the z axis is described
by
2
 2
ϕ θ ) − (ϕ θ ) + 1
(
M = ζωθ  d + 2
.
2
1 + (ζω K )


2
(2)
In the equation above, the two terms within the brackets represent contribution to the
rotational drag from the first and second links respectively. While the first link’s
contribution is constant, the second link’s contribution decreases with increasing ω as
the link bends inward towards the rotational (z-) axis, where the fluid flow is slower.
Similarly, for any given value of ω, the second-link’s contribution is lower for a
filament with higher intrinsic curvature φ. For φ = 0 straight filaments, and for d < 1 / 2,
the dependence of M on ω is shown in Fig. 3b.
From the equation, values of ω at the turning points are ω = t / 2 − 1 ± t / 2 1 − 8 / t where
2
2
t = (2 − φ θ ) + (φ θ ) d 2 . One can see from the determinant ∆ = 1 − 8 / t , that the
(
)
hysteresis loop closes when 8d 2 = (2 − φ θ ) + (φ θ ) . For d = 1 / 2 this corresponds to φ = θ.
At this point, the moment becomes a monotonic increasing function of angular velocity,
however a second order transition is still present at ω = t / 2 − 1 . The choice of d = 1 / 2 is
arbitrary, being the centre of drag of the lower half of the filament. For smaller d, this
model would never show complete disappearance of the hysteresis for any value of φ. In
contrast to the simulation results in Fig. 2, the final angular velocity of this basic model
does not decrease for increasing curvature, as it cannot capture the exact distribution of
forces along the length of the filament. However, this model is a good tool to understand
the origin of the transitions and that smoothing of the behaviour is indeed to be expected
in these kinds of filaments.
2
2
How will helicity affect thrust?
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S. Clark, R. Prabhakar
The other critical component that must be understood in order to use these filaments for
propulsion is the thrust. Linear thrust is possible for helical geometries under rotational
driving due coupling of the rotational and translational mobilities (Chwang, 1971). At
zero Reynolds number, this force is entirely a function of shape, which for these semiflexible filaments is itself result of complicated fluid-structure interactions. Returning to
our simulations, the thrust, or axial force on a filament is equal to the sum of the
NB
hydrodynamic forces on the filament in the z direction Fz = ẑ ⋅ ∑ µ =1 M −1µν g(drν dt ). Figure 4
shows the thrust forces generated by a selection of filaments with varying curvature.
This looks very similar to the trends of ω vs. T – strong hysteresis disappearing with
increasing curvature can be clearly identified. At low torques before the wrapping
transition, straight filaments ( ) produce very little thrust, which increases significantly
once the wrapped configuration is reached. On the other hand, filaments with intrinsic
twist see once again much smoother operating characteristics.
Within the range of torques investigated by this study, there is only a very small range at
which the straight filaments outperform the curved ones in terms of thrust produced.
The straight filament’s thrust appears to begin to plateau beyond an applied torque of T
~ 0.05, meaning that applying ever increasing torques may not have a strong effect on
any resultant motion. The filaments with higher curvature (●, κ = 1.42, , κ = 1) do not
have this plateau within the experimental range.
Fig. 4: Steady-state thrust as a function of applied torque in single filaments of different
intrinsic curvatures. Filaments of higher intrinsic curvature (●, κ = 1.42, , κ = 1)
produce higher thrust at low torques. Straight filaments ( , κ = 0) have a small range of
producing more thrust, but this is quickly lost. Dashed lines are drawn to guide the eye.
Finally we consider the effect of two co-twirled filaments. To look at this case we take
two filaments and drive them under the same torque at a fixed distance D from each
other. Under driving, these filaments are known to synchronise with one another and
intertwine (Berg 1974, MacNab 1977). This mechanism is often seen in nature, and
facilitates straight swimming. For straight filaments, a small degree of bundling is
possible (Fig. 5a) but it appears that it may not confer the same advantages. Figure 5b
shows the time series of the net thrust F z,tot =F z, 1 +F z,2 of both straight (green) and
curved (black) filaments. Unlike for isolated filaments, the periodic changes in filament
shape as a result of hydrodynamic interactions between filaments cause variation of the
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S. Clark, R. Prabhakar
thrust with time. These are very significant, for the straight filaments being up to 25% of
the average thrust.
Fig. 5: a) Straight filaments bundle and synchronise at high torques, although a tight
inter-wound bundle is not possible. b) Time variation of the total thrust force Fz,tot,
delivered by a pair of co-rotated filaments whose bases are separated by D = 0.6 (solid
curves). Dashed lines represent the total axial force for two filaments that are infinitely
far apart (i.e. twice the thrust of an isolated filament). Black curves are for high
curvature κ = 2.0, and green for low curvature κ = 0.1. The applied torque is T = 0.06.
This is in contrast to the more helical filaments where the variation is smaller. These
filaments hold their shape, and more uniform thrust results. The helical filaments
investigated here are only marginally twisted, since we want to understand how intrinsic
twist progressively changes behaviour. With helical filaments of one or more complete
turns, it is possible that the oscillations are completely eliminated. It is interesting to
note in both cases, that the combined thrust for a pair of proximate filaments when
bundled is less than that delivered by two isolated unbundled filaments. In other words,
the use of multiple helical flagella for bundling may not be a device to increase thrust,
but to achieve steady directional swimming. Real bacteria often possess more than two
flagella, and the effect of adding more flagella to the system still needs to be
investigated.
Conclusions
This study looks at the effect of helicity on a Stokesian semi-flexible filamentous
propeller. We used a toy model to help explain why intrinsic curvature will eliminate
sharp phase transitions and hysteresis and result in smoother speed-torque
characteristics. We show through simulations that thrust-torque characteristics are not
only smoother for helical filaments, but also that in most cases the axial force is higher
than for its straight counterpart. We also show that intrinsic helicity assists in giving
smoother thrust characteristics, both for single and bundled filaments pairs.
Bacterial flagellar filaments—which typically have two or more full helical turns—may
therefore experience no wrapping transitions and hence have smooth operating
characteristics, with no sharp changes in thrust with applied torque. Therefore, in both
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S. Clark, R. Prabhakar
singly- and multiply-flagellated organisms, filament helicity may confer significant
advantages to micro-swimmers.
ACKNOWLEDGEMENTS
This research was conducted thanks to the support of the Victorian Life Sciences
Computation Initiative (VLSCI), through the use of their Peak Computing Facility.
REFERENCES
Bekker, H., Berendsen, H. and Van Gunsteren, W. 1995, ‘Force and virial of torsionalangle- dependent potentials’, J. Comp. Chem., Vol 16, pp 527–533.
Berg, H. C. 1974, ‘Dynamic properties of bacterial flagellar motors’, Nature, Vol 249,
pp 77-79.
Chwang, A. 1971, Helical movements of flagellated propelling microorganisms, PhD
thesis, California Institute of Technology.
Clark, S. and Prabhakar, R. 2011, ‘Effect of helicity on wrapping and bundling of semiflexible filaments twirled in a viscous fluid’, Soft Matter, in press.
Macnab, R. 1977, ‘Bacterial flagella rotating in bundles: a study in helical geometry’,
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Manghi, M., Schlagberger, X. & Netz, R. 2006, ‘Propulsion with a rotating elastic
nanorod’, Phys. Rev. Lett. Vol 96, 68101.
Purcell, E. 1977, ‘Life at low Reynolds number’, Am. J. Phys Vol 45, pp 11.
Qian, B., Powers, T. & Breuer, K. (2008), ‘Shape transition and propulsive force of an
elastic rod rotating in a viscous fluid’, Phys. Rev. Lett. Vol 100, 78101.
Rotne, J. and Prager, S. 1969, ‘Variational treatment of hydrodynamic interaction in
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BRIEF BIOGRAPHY OF PRESENTER
Sarah Clark graduated from Monash University in 2007, with a double ScienceMechanical Engineering degree. She commenced her PhD under the supervision of Dr.
Prabhakar Ranganathan in 2009, and is funded by the Sir James McNeill and Finkel
Scholarships. She is currently the national student representative for the Australian
Society of Rheology.
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