Proceedings of the ASME 2012 Summer Bioengineering Conference
SBC2012
June 20-23, Fajardo, Puerto Rico, USA
SBC2012-XXXXXX
THE APPARENT FLEXURAL RIGIDITY OF THE FLAGELLAR AXONEME
DEPENDS ON RESISTANCE TO INTER-DOUBLET SLIDING
1
1
1
2
3
Gang Xu, Kate S. Wilson, Ruth J. Okamoto, Jin-Yu Shao, Susan K. Dutcher,
1,2
Philip V. Bayly
1
Department of Mechanical Engineering and Materials Science
2
Department of Biomedical Engineering
3
Department of Genetics
Washington University in Saint Louis
St. Louis, MO 63130
INTRODUCTION
Cilia are thin subcellular organelles that bend actively to propel fluid.
The ciliary cytoskeleton (the axoneme) consists of nine outer
microtubule doublets surrounding a central pair of singlet
microtubules. Large bending deformations of the axoneme involve
relative sliding of the outer doublets, driven by the motor protein
dynein. Ciliary structure and function have been studied extensively,
but details of the mechanics and coordination of the axoneme remain
unclear. In particular, dynein activity must be switched on and off at
specific times and locations to produce an oscillatory, propulsive beat.
Leading hypotheses assert that mechanical feedback plays a role in the
control of dynein activity, but these ideas remain speculative.
We use the alga Chlamydomonas reinhardtii to model mechanics
and control of the axoneme. The Chlamydomonas flagellum shares the
highly conserved structure of the ciliary axoneme, and is simpler to
study. The active forces of dynein motors can be determined from the
sum of the elastic and viscous forces on the flagellum. To exploit this
idea, we can measure the kinematics of the flagellar waveform [1].
Viscous forces are estimated from the local velocity of the flagellum
[2]. Elastic forces may be estimated from the shape of the flagellum
and its elastic resistance to bending. In addition to the stiffness of each
doublet, the resistance to relative sliding between doublets plays an
important role in the overall bending stiffness of the flagellum.
When the passive flagellum is bent by an external force at an
intermediate location, a counter-bend appears distal to the imposed
bend [3]. This behavior is consistent with a model in which doublets
are connected by elastic elements that produce a local shear force, ms
(bending moment per unit length) that depends on sliding
displacement. In a simple model, this force is proportional to the total
bend angle relative to the base, θ – θ0. Away from local internal or
external forces, the quasi-static shape of the flagellum is governed by:
EI
∂ 2θ
= m=
k (θ − θ 0 )
s
∂s 2
(1)
A bending test is used to find the “apparent flexural rigidity,”
EI . A counter-bend experiment is then used to identify the flexural
rigidity, EI, and shear stiffness, k. These coefficients were estimated in
Chlamydomonas wild type and mutant (pf3) cells lacking structural
components (nexin link) and a microtubule protein (p58) [4].
METHODS
Chlamydomonas wild type and pf3 cells were grown by standard
methods [5]. The apparent flexural rigidity of flagella was measured
with a custom-built optical tweezers system [6]. Stiffness was tested in
quiescent flagella, in which dynein activity was blocked by vanadate.
A cell held by micropipette suction was driven by piezoelectric stage
to approach and retract from a coated bead in the laser trap. Upon
contact, adhesion between the flagellar tip and the bead provided
piconewton-level tip force, bending the flagellum. Bead deflection and
cell body motion were measured with a single-particle tracking
algorism. Applied force is the product of the trap stiffness and bead
deflection. The deflection of the flagellar tip is the difference between
the displacements of the bead and cell body (Fig. 1). Division of force
(F) by the corresponding deflection of the flagellar tip (d) yields the
flexural stiffness of the flagellum (k = F/d), which was then used to
calculate the apparent flexural rigidity: EI = FL2 3d . The ratio
of elastic shear stiffness (resistance to interdoublet sliding) to flexural
rigidity was estimated by the counterbend response in bent flagella
manipulated by a glass microneedle (Fig. 2). This ratio was found by
fitting the shape of unloaded region to Eq. 3.
1
Copyright © 2012 by ASME
k s L2
s.
∂ 2θ
2
2
=
β
θ
=
β
=
,
where
, S
2
EI
L
∂S
(3)
Bending of the flagellum with a tip load is governed by
∂ 2θ
FL2
2
−
β
θ
=
−
δ ( S − 1) .
∂S 2
EI
(4)
For a fixed-free flagellum the maximum deflection at the tip is:
FL2
d=
EI
 1
1  1 − eβ
e− β − 1   .
+
 2 + 3

2β
1 + e −2 β  
β  1+ e
β
(5)
Figure 3. Shear stiffness and flexural rigidity of flagella. (a) The ratio
of the shear stiffness to flexural rigidity, found from the counterbend.
(b) The apparent flexural rigidity obtained from experiment (dots) and
Eq. 6 (lines). Averages are separated for long and short pf3 flagella.
Combining Equation 5 with the expression for EI for yields,
EI
3
3  1 − eβ
e−β −1  .

= 2 + 3 
+
2β
1 + e − 2 β 
β 1+ e
EI β
(6)
RESULTS AND DISCUSSION
The apparent flexural rigidity of wild-type and pf3 flagella is similar
(~2000-3000 pN·µm2) (Fig. 3b). This value is much greater than the
summed flexural rigidity of individual microtubules each bending with
respect to its own neutral axis, but not as large as if all microtubules
were bent around the central axis of the axoneme.
The ratio k/EI is shown in Fig. 3a, and flexural rigidity and interdoublet shear stiffness are given in Table 1. The EI value for wild-type
flagella closely approximates the sum of the EI values for individual
microtubules (5-10 pN-µm2). Surprisingly, the difference in EI is
relatively large between wild-type and pf3 flagella. The pf3 mutant is
known mostly for its lack of nexin links, which are expected to affect
shear stiffness. However, as noted by Yanagisawa and Kamiya [4], the
pf3 mutant is also deficient in p58, a homolog of the polypeptide
tektin, that may affect the properties of the doublet microtubules.
The most important conclusion of this study is that the flexural
rigidity of the flagellum, and the inter-doublet shear stiffness can be
estimated by a combination of bending experiments and used to gain
insight into the mechanism of flagellar and ciliary oscillation.
(d)
L(µm)
EI (pN·µm2)
EI (pN·µm2)
k s (pN/rad)
WT
7.6±1.5
2700±1100 (n=23)
238
117
pf3
8.8±0.9
2900±1500 (n=21)
102
101
Table 1. Summary of elastic constants for wild type and pf3 flagella.
ACKNOWLEDGMENT: G. Xu acknowledges support from a
Children’s Discovery Institute Postdoctoral Fellowship.
2nd derivative of angle (rad/um2)
Figure 1. Bending the flagellum with optical trap. Video
micrographs (a-c) and schematics (a’-c’) show the flagellum
approach, push, and pull on a trapped bead (springs represent
trap stiffness). (d) Typical tracking curves for cell body or
pipettte (red) and bead (green) positions (two pulling cycles).
REFERENCES
1. Bayly, P.V., et al., Efficient spatiotemporal analysis of the
flagellar waveform of Chlamydomonas reinhardtii.
Cytoskeleton (Hoboken), 2010. 67(1): 56-69.
2. Bayly, P.V., et al., Propulsive forces on the flagellum during
locomotion of Chlamydomonas reinhardtii. Biophys J, 2011.
100(11): 2716-25.
3. Pelle, D.W., et al., Mechanical properties of the passive sea
urchin sperm flagellum. Cell Motil Cytoskeleton, 2009.
66(9): 721-35.
4. Yanagisawa, H.A. and R. Kamiya, A tektin homologue is
decreased in chlamydomonas mutants lacking an axonemal
inner-arm dynein. Mol Biol Cell, 2004. 15(5): 2105-15.
5. Dutcher, S.K., Mating and tetrad analysis in Chlamydomonas
reinhardtii. Methods Cell Biol, 1995. 47: 531-40.
6. Ying, J., et al., Unfolding the A2 domain of von Willebrand
factor with the optical trap. Biophys J, 2010. 98(8):1685-93.
0
data
linear fit
-0.2
-0.4
y = - 0.32*x - 0.031
-0.6
-0.8
0
0.5
1
Angle (rad)
1.5
2
Figure 2. The counterbend response of the flagellum indicates the
resistance to inter-doublet shear. (a-c) Video micrographs of a
flagellum manipulated with a glass microprobe. The cell body is
held by micropipette suction. A bend of the flagellum is induced by
the probe close to the base and is accompanied by a counterbend
distal to the probe. (d) A linear fit to the plot of second derivative of
θ vs θ yields the ratio of the shear stiffness to the flexural rigidity.
2
Copyright © 2012 by ASME