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First published online June 15, 2007
Journal of Experimental Biology 210, 2383-2389 (2007)
Published by The Company of Biologists 2007
doi: 10.1242/jeb.004572
Mechanosensation and mechanical load modulate the locomotory gait of swimming C. elegans
1 Department of Physics, Harvard University, Cambridge, MA 02138,
USA
2 Division of Engineering and Applied Sciences, Harvard University,
Cambridge, MA 02138, USA
3 Department of Organismic and Evolutionary Biology, Harvard University,
Cambridge, MA 02138, USA
* Author for correspondence (e-mail: samuel{at}physics.harvard.edu)
Accepted 25 April 2007
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Summary |
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 TOP Summary IntroductionMaterials and methodsResultsDiscussionReferences |
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Key words: nematode, locomotion, mechanosensation, Caenorhabditis elegans
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Introduction |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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). While much of this work was motivated by and applied to the
movements of microscopic swimmers such as spermatozoa and bacteria, it is just
as applicable to larger swimmers moving in a hydrodynamic environment where
inertia is dominated by viscous forces, i.e. where the Reynolds number is less
than unity. However, the connection between external kinematics and the
internal sensorimotor circuits that govern the adaptability of gait in
different environments remains largely unexplored.
The nematode Caenorhabditis elegans provides a model system that
allows rigorous analyses of locomotory gait at multiple levels. One advantage
is anatomical simplicity. The C. elegans nervous system contains
precisely 302 neurons with a stereotyped wiring diagram, and undulating
propulsion is driven by rhythmic activity of 95 muscle cells that line its
dorsal and ventral sides (White et al.,
1976; White et al.,
1986). Another advantage is straightforward hydrodynamics, as the
Reynolds numbers of swimming C. elegans are less than unity
(Purcell, 1977). Finally,
C. elegans offers genetic accessibility, facilitating an analysis of
locomotory gait that traverses the levels of genetics, neurophysiology and
behavior.
Most studies of C. elegans behavior involve crawling motility on
agar surfaces. C. elegans improves its own crawling efficiency on
soft agar surfaces by incising a sinusoidal groove, thereby reducing lateral
slipping of its body against the groove. Moreover, crawling worms use
undulations with shorter wavelength and slower temporal frequency than
swimming worms. These differences underscore the adaptability of locomotory
gait in response to the physical nature of the external medium. However,
crawling locomotion is particularly complicated when analyzing external
kinematics: in principle, one has to take into account the surface tension
that holds the worm to the agar surface, the forces that incise the groove,
the variable friction and tension along the length and around the perimeter of
the worm's cylindrical body, and so on. Thus, Gray and Lissman focused on
swimming locomotion in their classic analysis relating the speed of
progression of nematodes to the form and frequency of their undulating waves
(Gray and Lissman, 1964).
Using C. elegans, it may be possible to connect the adaptability
of locomotory gait to the genetics and physiological operation of sensory and
motor systems. In a recent analysis, Karbowski et al. showed that the spatial
form and temporal frequency of the crawling gait is largely independent of
external physical parameters, including worm size and the rigidity of the agar
surface, and that the crawling gait is unaffected by many mutations that
affect neurons and muscle (Karbowski et
al., 2006). At the molecular level, Tavernarakis et al. showed
that mutation of the mec-6 or unc-8 genes, which
encode subunits of mechanically gated ion channels, leads to an irregular
crawling gait with reduced amplitude and wavelength
(Tavernarakis et al., 1997),
suggesting that mechanosensory systems may play an important role in
locomotory gait. Li et al. discovered that mutation of the trp-4
gene, which encodes a mechanosensitive TRPN channel expressed in the DVA
interneuron, leads to a crawling gait with increased amplitude and speed
(Li et al., 2006), thus
identifying a molecular and neuronal pathway for proprioception in regulating
the crawling gait.
Here, we focus on the adaptability of locomotory gait for the more
tractable case of swimming C. elegans. The movement of slender
objects through viscous fluids at low Reynolds numbers is well understood,
allowing us, like Gray and Lissmann (Gray
and Lissmann, 1964), to precisely calculate the flow field
surrounding the swimming worm, the normal and transverse forces at each point
on the worm's surface and thus the speed and efficiency of undulating
propulsion. By analyzing the movements of individual worms swimming in fluids
containing different amounts of the viscous agent methylcellulose, we
characterize the effects of increasing mechanical load on the forward swimming
gait. We also characterize the contribution of mechanosensory feedback to the
swimming gait by quantifying the effects of mechanosensory mutation and laser
ablation of touch receptor neurons. Our observations lay groundwork for
analyzing the effects of mechanical load and mechanosensory input in the
adaptability of locomotory gait for swimming C. elegans.
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Materials and methods |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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). The mec-4(d) mutant strain was obtained from
Monica Driscoll (Rutgers, Piscataway, NJ, USA) and the mec-6(u450)
mutant strain was obtained from the C. elegans Genetics Center
(Minneapolis, MN, USA).
Laser ablation
We followed techniques established for standard laser microsurgery
(Bargmann and Avery, 1995),
except that we used an ultrafast Ti:sapphire laser to generate the laser
pulses (Chung et al., 2006). To
facilitate the identification and killing of ALM and PLM neurons in L1 worms,
we used transgenic worms that expressed the mec-4::gfp
transgene in those neurons. Each larval animal was anesthetized with 0.2-0.8
mmol l-1 sodium azide and mounted on a thin pad of 2% agarose.
Using fluorescence microscopy, each neuronal target was moved to the focal
point of laser pulses and ablated. After surgery, worms were recovered within
30 min by placing them on fresh plates with bacterial food. After recovery,
worms were returned to overnight incubation before undergoing behavioral
assays as young adults.
Methylcellulose
Stock solutions of viscous fluid were prepared by adding certain amounts of
methylcellulose (Sigma, St Louis, MO, USA) to NGM buffer
(Sulston and Hodgkin, 1988)
and stirring overnight at 4°C. We determined the viscosity of each
solution by quantifying the sedimentation rate of small beads dropped into
each fluid. Methylcellulose is a polymeric material that behaves like a
viscoelastic liquid at high strain rates but exhibits Newtonian hydrodynamics
at sufficiently low shear rates. The typical shear rates exhibited by swimming
C. elegans in our experiments were 
15 s-1. In order
to show that non-Newtonian effects were negligible for the shear rates
exhibited by swimming C. elegans in our experiments, we used
sedimentation experiments to verify the transverse and longitudinal frictional
drag coefficients of thin metal rods with similar size and aspect ratio of
adult C. elegans.
Video analysis
Individual young adult worms were removed from overnight cultivation at
20°C, rinsed in NGM buffer and immersed in a viscous solution made with
NGM buffer and defined concentrations of methylcellulose. The worm was
obliquely illuminated to render the worm's body in sharp contrast to a dark
background. A CCD camera equipped with a zoom lens was used to capture images
of the swimming worms at 30 Hz, and 10 s movie clips were recorded using
LabVIEW (National Instruments, Austin, TX, USA).
We used our own machine-vision algorithms written in Matlab (MathWorks,
Inc., Natick, MA, USA) to analyze those movie clips in which the worm was
observed to be swimming within the focal plane and within the bulk liquid. We
restricted our analysis to experiments in which the 0.1 mm-diameter worm was
at least 5 mm away from the top and bottom surfaces of the liquid, where
hydrodynamic corrections due to viscous coupling with the fluid surfaces would
be <2% (Brenner, 1961). An
edge-detection algorithm identified the x-y coordinates of
all points along the dorsal and ventral sides of the swimming worm. The dorsal
and ventral edges were least-squares fit with a Bezier spline (degree 5, 12
control points, open uniform knot), rendering them continuous and
differentiable (Cohen et al.,
2001). Centerline points were calculated at the bisection of the
vector connecting each point on the dorsal side to the nearest point on the
ventral side. These midline points were then also fit to a Bezier spline and
the coordinates of the body centerline were calculated at N=100
evenly spaced points from nose to tail (see Movie in supplementary material).
The result was a continuous and smoothly differentiable mathematical
description of the coordinates of the body centerline at each point in time -
[xi(t), yi(t)] from
i=1 to N - simplifying subsequent numerical analysis of gait
dynamics.
We used the coordinates of the body centerline to calculate the curvature
as a function of body coordinate and of time, as well as the instantaneous
velocity of each point along the body centerline. We calculated the curvature
at each point along the body centerline within each video frame using the
N-1 tangent angles along the centerline:
i=arctan[(yi-yi-1)/(xi-xi-1)].
We calculated the velocity at each point along the body centerline using the
coordinates from successive video frames:
ui=[xi(t+
t)-xi(t),yi(t+t)-yi(t)]/t.
These calculated velocities were used to determine the longitudinal and
normal forces along the body centerline and swimming power using standard
methods of resistive-force theory (Gray
and Hancock, 1955).
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Results |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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). We
developed a similar algorithm that tracks every point along the worm
centerline afforded by video resolution and renders the worm centerline as a
continuous, differentiable two-dimensional space curve. The instantaneous
shape of the worm may be fully described by defining the curvature (the
inverse of the radius of curvature) at each point along the worm's body
(Fig. 1A) (see Movie in
supplementary material). We define a body coordinate (l) along the
body centerline along the total length from head (l=0) to tail
(l=L). In terms of the tangential angle to each point along the
centerline (
) and the body coordinate, the curvature (
) at each
point is defined by:
 | (1) |
(l,t)
(Fig. 1B).
|
In order to characterize the energetics and efficiency of the swimming
gait, we require the time-varying hydrodynamic force against each segment of
the worm's body within each undulation cycle. C. elegans is a
microscopic, elongate swimmer about 1 mm in length (L) and 0.1 mm in
diameter (d), allowing us to combine fluid dynamics at low Reynolds
numbers and slender-body approximations in making these calculations. In the
domain of low Reynolds number hydrodynamics, force is linearly proportional to
velocity. Using video analysis, it is straightforward to quantify the
instantaneous velocity of each point along the body centerline
(Fig. 1C). To characterize the
normal and tangential hydrodynamic force along the body centerline, we only
need to multiply these velocities by frictional drag coefficients,
CN and CL, associated with normal and
longitudinal movements of slender objects
(Berg, 1993). For example, for
a prolate ellipsoid with length L and diameter d, these
frictional drag coefficients in a fluid with viscosity 
are:
 | (2) |
Mechanical load reduces the frequency but does not change the geometric shape of the swimming gait
Next, we quantified the effect of mechanical load on the shape and temporal
dynamics of the swimming gait. We immersed individual young adult worms in
solutions containing different amounts of the viscous agent methylcellulose,
ranging from 0.5% methylcellulose at 0.05 Pa s to 3% methylcellulose at 50 Pa
s (Berg and Turner, 1979), and
monitored their forward-swimming movements using video microscopy. We found
that the shape of the swimming gait characterized by its undulation wavelength
is remarkably invariant across this 1000-fold increase in external mechanical
load (Fig. 2A). However, we
found that the undulation frequency exhibits a shallow dependence on
mechanical load, dropping from about 2 Hz at 0.05 Pa s to 0.5 Hz at 50 Pa s
(Fig. 2B).
|
), a
lower limit on the total power that the worm uses to swim, was obtained by
integrating the products of force and velocity along the body centerline and
averaging over an undulation cycle:
 | (3) |
is the undulation period, uL and
uN are the longitudinal and normal velocities,
respectively, l is length along the body and t is time. The
dependence of the viscous power dissipation rate and viscosity (over the range
of the viscosity where we were able to examine swimming worms) shown in
Fig. 2C suggests that, in this
regime, the worm is not power-limited either.
Removing mechanosensory input increases the frequency of the swimming gait
Effects of mechanical load on the swimming gait may appear at multiple
levels, from mechanosensory input to the contractile mechanisms of muscle
cells (Duysens et al., 2000).
Mechanical load (the external hydrodynamic force on the swimming worm) and
mechanosensation (the worm's ability to detect external forces) must be
coupled at the level of external kinematics. In C. elegans, the genes
and neurons that mediate mechanosensation have been extensively analyzed,
providing us with an opportunity to dissect the specific contribution of
mechanosensory input to the regulation of the swimming gait
(Goodman and Schwarz, 2003).
When the worm is gently touched on the head (tail), it responds by crawling
backward (forward). The worm has six touch receptor neurons, also called
microtubule cells, that mediate these touch responses. By laser ablating the
touch receptor neurons, Chalfie et al. showed that either the ALML or ALMR
neuron (the left and right anterior lateral microtubule cells) is required for
the full response to touch on the head, that either the PLML or PLMR neuron
(the left and right posterior lateral microtubule cells) is required for the
full response to touch on the tail, that the AVM neuron (the anterior ventral
microtubule cell) has only a weak contribution to touch responses to the head,
and that the PVM neuron (the posterior ventral microtubule cell) does not
contribute to touch responses (Chalfie et
al., 1985).
Touch sensation by all of the touch receptor neurons may be altered by
mutations in the mec-4 and mec-6 genes, which encode
membrane proteins within the mechanosensory ion channel complex
(Chelur et al., 2002;
O'Hagan et al., 2005). The
mec-4(d) mutation leads to actual degeneration of the touch cells,
thereby abolishing the touch response
(Driscoll and Chalfie, 1991).
The mec-6(u450) mutation interferes with the operation of
mechanosensitive ion channels within the touch cells and also abolishes the
touch response (Chelur et al.,
2002). We examined both mec-4(d) and mec-6(u450)
mutants in viscous fluids and found that neither mutation affects the shape of
the swimming gait; both mutants exhibit the same undulation wavelength as
wild-type worms in viscous fluids ranging from 0.05-50 Pa s
(Fig. 3A). However, both
mutations significantly affect the undulation frequency
(Fig. 3B). Surprisingly, both
mec-4(d) and mec-6(u450) mutant worms exhibit higher
undulation frequencies than wild-type worms at each value of mechanical load.
However, increasing mechanical load systematically reduces the undulation
frequency of mec-4(d) and mec-6(u450) worms at nearly the
same rate as wild-type worms.
|
We sought additional evidence that the touch receptor neurons are involved in regulating the swimming gait through laser ablation analysis. We used transgenic animals that expressed the mec-4::gfp transgene in a wild-type background, allowing us to target and laser ablate the ALM and PLM touch receptor neurons in larval animals. We found that ablating the ALM touch neurons has no effect on the shape of the swimming gait (Fig. 3C) but has the same effect on the undulation frequency as either the mec-4(d) or mec-6(u450) mutations (Fig. 3D). Ablating the PLM neurons has no effect on the shape of the swimming gait (Fig. 3C) and a lesser effect on the undulation frequency (Fig. 3D). Ablating both the ALM and PLM neurons was indistinguishable from killing ALM alone. These observations agree with the mutant analysis and suggest that much of the mechanosensory input that regulates the forward-swimming gait derives from the activity of the ALM touch receptor neurons.
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Discussion |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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). Here, we discuss several implications of our results on the
neuromuscular mechanisms and biomechanics that underlie the worm's swimming
gait.
The neuromuscular circuit in C. elegans that generates and
maintains its rhythmic pattern of locomotory gait is not yet understood
(Syntichaki and Tavernakis,
2004). In a central pattern generator (CPG) model proposed by
Niebur and Erdos (Niebur and Erdos,
1993), a circuit in the nerve ring composed of command
interneurons (AVA, AVB, AVD, and PVC) maintains rhythmic activity through
synaptic interconnections within the circuit. This CPG then drives undulations
through synaptic connections along the ventral nerve cord between the command
interneurons and the motor neurons. Interestingly, these command interneurons
are postsynaptic to the touch receptor neurons and are directly involved in
the touch response. Chalfie et al., using systematic laser ablation analysis,
showed that the ALM touch receptor neurons mediate most of the touch response
to the head by stimulating AVB and PVC, which trigger reverse movement, and
inhibiting AVA and AVD, which trigger forward movement
(Chalfie et al., 1985). They
also showed that the PLM touch receptor neurons mediate the touch response to
the tail by having precisely opposite effects on the command interneurons
(Chalfie et al., 1985). One
possibility is that the touch receptor neurons, in addition to mediating the
reflexive touch responses, also affect the frequency of the swimming gait by
altering the temporal dynamics of the CPG, a direct consequence of their
rhythmic activation by the external forces caused by movement. Thus, removing
the ALM neurons, which trigger reverse movement and conceivably have an
antagonistic effect on forward movement, might increase the temporal frequency
of the swimming gait. However, it is unclear why removing the PLM neurons,
which presumably have an opposite relationship with the command interneurons,
also increases the speed of forward movement, albeit by a lesser amount. These
surprising observations may be clarified by better dynamical models of the CPG
circuits that incorporate the effects of mechanosensory input and the worm's
self-movement in its surroundings.
Our observations support basic conclusions about the biomechanics of
self-propulsion in C. elegans. One possibility is that the forces
required to displace the surrounding fluid as the worm swims are negligible in
comparison to the forces required to bend its rigid body. In any case, C.
elegans is capable of generating much higher propulsive forces under
mechanical load than in water. In other words, C. elegans effectively
swims in low gear, which might have direct ecological relevance, enabling the
animal to burrow through soil, a natural habitat where it likely encounters
higher mechanical loads than when swimming in water. Another striking
observation is that the shape of the swimming gait is invariant over 1000-fold
changes in mechanical load. When swimming, the worm has only about one
wavelength of undulation along its body, whereas when crawling it has closer
to two (Karbowski et al.,
2006). One possibility is that the variation in the shape of the
locomotory gait is a simple function of mechanical load. We originally thought
that swimming in water might represent the low end of mechanical load
(characterized by a swimming gait with large wavelength and high temporal
frequency) and crawling on agar surfaces might represent the high end of
mechanical load (characterized by small wavelength and low temporal
frequency), and that by progressively varying mechanical load through
viscosity we might uncover a gradual transformation in the spatiotemporal
dynamics of gait with intervening wavelengths and frequencies. Gait may still
be a straightforward function of mechanical load: the mechanical load of the
most viscous aqueous solution that we were able to make with methylcellulose
(50 Pa s at 3% w/w) is probably lower than that required to crawl on agar
surfaces, and the transition between the swimming and crawling gaits may be
more subtle. For example, eelworm larvae exhibit a transition from a crawling
to swimming gait that may be evoked by gradually increasing the film thickness
of water on a substrate on which the eelworm larvae crawl
(Wallace, 1959).
It may be possible, however, that the swimming and crawling gaits are
well-adapted solutions to fundamental differences in the mechanics of swimming
at low Reynolds numbers and crawling on agar surfaces. When crawling, C.
elegans incises ridges into the agar surface with its body, pushing
laterally and sliding longitudinally along these ridges. This strategy
improves crawling efficiency by making the frictional drag coefficient for
normal movement (CN) much higher than the frictional drag
coefficient for longitudinal movement (CL). However, when
swimming at low Reynolds number in Newtonian fluids, the 1:2 ratio between
CN and CL is fixed. Gray and Hancock
showed that the calculated speed of a slender object propelling itself by
undulating waves at low Reynolds numbers depends on the relative values of
CN and CL, which are prescribed by
hydrodynamics (Gray and Hancock,
1955). Holwill and Burge
(Holwill and Burge, 1963)
extended the analysis of such swimmers to include considerations of energetics
and efficiency. They found that the hydrodynamic efficiency - defined as the
ratio between the power that the undulatory swimmer would use to move at a
particular speed and the power that an external agent would use to pull the
swimmer passively at the same speed, a quantity that is independent of the
viscosity of the fluid for low Reynolds number movements - reaches a peak when
the swimmer uses one undulation wavelength over its entire body. Thus, C.
elegans, although it can change the shape of its locomotory gait, may opt
to use the same spatial form in various Newtonian fluids in order to maximize
hydrodynamic efficiency. In order to explore the transformation of gait in
different environments, it may be useful to study the movements of C.
elegans in environments where the ratio between CL
and CN may be varied systematically.
C. elegans provides a rare opportunity to analyze locomotory gait
that connects our understanding of the genetics and neurobiology of
mechanosensory transduction (Goodman and
Schwarz, 2003) at the molecular level with the mechanics of
low-Reynolds number hydrodynamics at the organismal level
(Purcell, 1977). Here, we have
taken advantage of these tools to isolate and quantify specific effects of
mechanosensory and mechanical load on the swimming gait. Our analysis
highlights new questions: C. elegans opts to use the same spatial
form when swimming in fluids with different viscosity, but does it do so
because only certain spatial forms of locomotory gait are hardwired into its
motor systems or because the worm is using biochemical feedback to maximize
hydrodynamic efficiency? Mechanosensory input affects the temporal frequency
of the swimming gait, but why and how does this happen without affecting the
spatial form of the swimming gait? In any case, our analysis of the physical
and sensory determinants of the swimming gait provides a framework for
continued investigation of such questions within a tractable model system.
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Acknowledgments |
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Footnotes |
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as a
fraction of the total length of the body centerline. (B) The temporal
frequency of the undulating gait of swimming worms drops with increasing
mechanical load. (C) The mean power of the swimming gait calculated using
