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First published online November 17, 2006
Journal of Experimental Biology 209, 4652-4662 (2006)
Published by The Company of Biologists 2006
doi: 10.1242/jeb.02590
Sensorimotor control during isothermal tracking in Caenorhabditis elegans
1 Department of Physics, Harvard University, Cambridge, MA 02138,
USA
2 Department of Biology, Brandeis University, Waltham, MA 02453,
USA
3 Division of Engineering and Applied Sciences, Harvard University,
Cambridge, MA 02138, USA
4 Department of Organismic and Evolutionary Biology, Harvard University,
Cambridge, MA 02138, USA
Authors for correspondence (e-mail:
lm{at}deas.harvard.edu;
samuel{at}physics.harvard.edu)
Accepted 3 October 2006
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Summary |
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Key words: C. elegans, thermotaxis, sensorimotor control
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Introduction |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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). In order to be useful, descriptive models of behavior must
compactly and accurately represent the animal's behavior in terms of distinct
input-output relationships that facilitate mapping to the basic operations of
neuronal circuits.
The best understood sensorimotor behaviors in C. elegans are
reflexive avoidance to body touch (Chalfie
et al., 1985), its chemotactic ability to crawl up and down
chemical gradients (Bargmann and Horvitz,
1991), and its thermotactic ability to descend thermal gradients
in what is called cryophilic movement
(Hedgecock and Russell, 1975).
When C. elegans is touched on its head or tail, it reflexively crawls
backward or forward with propulsive undulations
(Chalfie et al., 1985). When
the animal crawls up or down chemical or thermal gradients, it tends to extend
(shorten) periods of forward movement by reducing (increasing) the stochastic
occurrence of reorientation maneuvers in response to improving (declining)
conditions. This biased-random walk strategy leads to net migration towards
favored environments (Pierce-Shimomura et
al., 1999; Ryu and Samuel,
2002). The systematic analysis of neuronal lesions in C.
elegans that disrupt touch avoidance or change the occurrence of
reorientation maneuvers has uncovered parts of the C. elegans nervous
system that contribute to the sensorimotor transformations of their respective
behaviors (Chalfie et al.,
1985; Tsalik and Hobert,
2003; Wakabayashi et al.,
2004; Gray et al.,
2005).
C. elegans also has the ability to track isotherms in spatial
thermal gradients when it navigates at temperatures near its previous
cultivation temperature (Hedgecock and
Russell, 1975). Although isothermal tracking behavior in C.
elegans has been known for 30 years, the sensorimotor strategy that
allows the worm to actively maintain isothermal alignment in spatial thermal
gradients has not been characterized. Unlike the biased random walk strategy
that characterizes cryophilic movement down thermal gradients, the isothermal
tracking behavior is strikingly deterministic: an individual animal strays
from each isothermal track by <0.1°C over prolonged periods of forward
movement (Hedgecock and Russell,
1975; Ryu and Samuel,
2002). Although the isothermal tracking behavior is often used as
a metric of thermotactic memory (e.g.
Gomez et al., 2001;
Biron et al., 2006), how C.
elegans transforms thermosensory perception of the surrounding thermal
gradient into deterministic patterning of its own movements is not known.
Here, we characterize sensorimotor control in isothermal tracking by
quantifying the trajectories of individual animals navigating defined
spatiotemporal thermal gradients, and offer a minimal mathematical model of
the sensorimotor transformation at the behavioral level that is consistent
with experimental observations.
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Materials and methods |
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).
Isothermal tracking assays
In each assay, 20-25 young adult worms were washed in NGM buffer for 
1
min before being moved to a 9-cm NGM agar plate
(Sulston and Hodgkin, 1988).
The plate was then placed on an aluminum surface under thermoelectric control
(Ryu and Samuel, 2002). Linear
thermal gradients were established by separately controlling the temperature
at the ends of a rectangular aluminum surface. Radial thermal gradients were
established by separately controlling the temperature of a copper pin at the
center of an aluminum disk and the temperature of the disk's perimeter. Worms
were illuminated by a ring of superbright LEDs. Video was captured using a CCD
camera for 30 min at 0.5 frames s-1 using LabVIEW (National
Instruments, Austin, TX, USA). We used low magnification (10 cm across a video
frame) to characterize the trajectories of individual animals on spatial
thermal gradients and at higher magnification (2 cm across a video frame) to
quantify the wavelength, speed and undulation frequency during isothermal
tracking. Video was analyzed using particle-tracking and analysis algorithms
written in MATLAB (Mathworks, Natick, MA, USA)
(Crocker and Grier, 1996).
Separate custom-written software for population-wide behavioral quantification
enabled us to simultaneously monitor and analyze the movements of 20-40 worms
navigating each spatial thermal gradient. In order to detect isothermal
tracks, the software decomposed the trajectory of each worm into runs and
turns, and scored each track as a run deviating from isothermal alignment by
less than <0.1°C.
Superposed temporal variation
Agar surfaces were heated and cooled in a spatially uniform manner by
feedback controlled illumination from a heat lamp held 1 m above the surface.
Temperature was monitored at the surface of the plate using a small
thermocouple (0.2 mm, Physitemp Instruments, Clifton, NJ, USA), which then
regulated voltage to the heat lamp with a PID feedback loop operating on a
computer running LabVIEW (National Instruments).
Simulations
Numerical simulations of the model represented by Eqn 2 and Eqn 3 were
carried out using standard integration packages in MATLAB.
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Results |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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). When tracking a specific isotherm, C. elegans
deviates from that isotherm by about 0.1°C. We quantified the percentage
of time that an individual worm spends tracking isotherms by quantifying the
total duration of all isothermal movements exhibited by populations of C.
elegans navigating gradients of defined steepness in 30 min. Thus, we
found that on gradients >0.2°C cm-1 in steepness, C.
elegans spends 15-20% of its time tracking isotherms, and the rest
of its time moving randomly in other directions in the intervals preceding and
following each isothermal track (Fig.
1C). The tendency to move isothermally is reduced on spatial
thermal gradients shallower than 0.1°C cm-1, which appears
to represent the threshold steepness for the behavior
(Fig. 1C).
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The ttx-3 and ttx-4 mutations cause C. elegans
to accumulate at the coldest points and warmest points on spatial thermal
gradients, and are called cryophilic and thermophilic phenotypes,
respectively. The ttx-3 gene encodes a LIM homeobox gene that is
required for normal development of the AIY interneuron
(Hobert et al., 1997). The
ttx-4 gene encodes a protein kinase C that is broadly expressed in
C. elegans sensory neurons, including the AFD thermosensory neuron
(Okochi et al., 2005). We
found that both mutations severely disrupt the ability of C. elegans
to track isotherms near the temperature of its previous cultivation
(Fig. 1F). This observation
suggests that isothermal tracking behavior may be particularly fragile with
respect to genetic mutations that affect thermosensation, as in order for
C. elegans to track isotherms for extended periods of time, the worm
must be able to detect both positive and negative temperature fluctuations
from isothermal alignment with exquisite sensitivity and reliability.
Isothermal tracking occurs in the presence of bacterial food and is unaffected by starvation
Since C. elegans only tracks isotherms near its previous
cultivation temperature, this form of long-term experience-dependent
plasticity is often assumed to reflect an associative link between the
presence of bacterial food and a particular temperature
(Hedgecock and Russell, 1975;
Mohri et al., 2005). Recently,
we found that this form of thermotactic plasticity does not, in fact, require
food; sustained exposure to a new temperature without food leads to isothermal
tracking near the new temperature (Biron et
al., 2006). In this study, we found that C. elegans also
tracks isotherms in the presence of bacterial food
(Fig. 2A). The average duration
of isothermal tracks of C. elegans crawling on spatial thermal
gradients in bacterial lawns is actually longer than that in the absence of
food (Fig. 2B). In order to
test whether starvation affects the ability to track isotherms, we cultivated
C. elegans at 20°C, and then starved them at 20°C for fixed
periods of time. We then allowed these starved worms to navigate spatial
thermal gradients near 20°C, and counted the number of isothermal tracks
that they exhibited in a fixed interval of time. We found no significant
variation in the number of isothermal tracks exhibited after starvation for as
long as 8 h (Fig. 2C). We
conclude that the presence or absence of food has limited relevance to
isothermal tracking. Instead, isothermal tracking might simply be a
homeostatic mechanism; moving to new temperatures is likely to require
adaptation in the rates of numerous biochemical and metabolic pathways, an
expense that can be avoided by tracking isotherms.
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Isothermal tracks are not preceded by deterministic movement
Isothermal tracks are usually preceded and followed by long periods of
undirected movement, in which successive runs fail to be oriented along
isotherms. We asked whether the movement of individual animals in the time
interval preceding each isothermal track is stereotyped, which would provide
evidence for deterministic pursuit of isothermal alignment. We quantified the
direction and duration of the preceding runs of each isothermal track
exhibited by C. elegans navigating a linear thermal gradient
(Fig. 3A). We found that
neither the direction nor the duration of the preceding runs are correlated
with the direction of the subsequent track, although the size of the
reorientation maneuver that separates the preceding run from the subsequent
isothermal track tends to be <90°
(Fig. 3B). Although isothermal
alignment appears to be initially serendipitous, the alignment is thereafter
actively maintained, as exhibited in the radial gradient assays. In rare but
striking cases, worms terminate one isothermal track and rapidly begin
tracking a different isotherm; the intervening time interval between two
different isothermal tracks can be as little as 10 s
(Fig. 3C). Therefore,
isothermal tracking does not appear to reflect commitment to, or long-term
memory of, the absolute temperature of each isotherm. Instead, it appears that
each isothermal track is begun anew when a reorientation maneuver
serendipitously provides isothermal alignment. The strategy for isothermal
tracking allows the worm to maintain isothermal alignment once it is found,
but not to pursue isothermal alignment. This observation also suggests that
the long-term memory of cultivation temperature is actually represented by the
range of temperatures in which a worm may track isotherms (e.g. the
distribution depicted in Fig.
1B), not by the specific temperatures of each isotherm that an
individual might track.
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Movements of C. elegans in spatial gradients with steep superposed temporal gradients
Thermosensory input in C. elegans is likely to occur at the front
of its head, at the sensory endings of neuronal dendrites like those of the
AFD thermosensory neuron (Perkins et al.,
1986; Cassata et al.,
2000; Satterlee et al.,
2001; Clark et al.,
2006). When C. elegans crawls in a spatial thermal
gradient, a thermoreceptor at the head measures an oscillating thermosensory
input driven by the worm's undulating gait and side-to-side movements.
Isothermal alignment in a spatial thermal gradient then implies that the
thermosensory input will oscillate with warming and cooling phases that are
equal and opposite, and the time-derivative of the measured temperature
changes will be 
T v, where T is the steepness of
the ambient spatial gradient and v is the instantaneous velocity of
the worm's head. The simplest strategy that utilizes temporal sensory
processing to maintain isothermal alignment is for the worm to actively strive
to balance the warming and cooling phases of the thermosensory input. Indeed,
imposing temporal variations on the environment that are comparable to or
exceed the temporal variations driven by self-movement in the spatial gradient
allows us to test if such a strategy is at work.
In the experiments shown in Fig.
5A,B, the maximum rate of temperature change imposed on the
environment (
T00.005°C s-1,
where is the imposed frequency and T0 is the
imposed amplitude) is less than the maximum rate of temperature change driven
by self-movement of the head on those spatial gradients
(|v||T0.012°C s-1),
which allows the worm to follow the moving isotherms in stable floral
trajectories. In Fig. 5C, we
show the results of an increase in the rate of imposed temperature variation
to T00.010°C s-1, which
disrupts the floral trajectories as they begin to loop back on themselves.
Note that these loops are still approximately isotherms. In
Fig. 5D, we show that
increasing the steepness of the spatial gradient restores the floral
trajectories by increasing the temporal variations driven by self-movement. We
quantified the relative occurrence of floral trajectories and looping
trajectories of worms navigating spatial gradients with 0.4°C
cm-1 steepness and with sine-wave temporal variations with
different periods and amplitudes. We found that the boundary between the two
types of trajectories in terms of the parameters of the spatiotemporal thermal
gradients occurs when T0 and
|v||T are comparable, which is consistent
with a model of temporal processing of thermosensory input.
Although floral pattern trajectories and looping trajectories are visibly distinguishable, they may be different instantiations of the same sensorimotor transformation, executed in different parameter spaces of the spatiotemporal thermal gradient. During either type of trajectory, C. elegans actively steers the direction of its forward movement in response to thermosensory input. The absence of abrupt reorientation maneuvers during isothermal tracking suggests that the worm must be actively and continuously modulating the curvature of its propulsive undulations during isothermal tracking. When C. elegans has perfect isothermal alignment, thermosensory input at the head will have perfectly balanced warming and cooling phases. However, any deviation from isothermal alignment will produce an asymmetry in the warming and cooling phases of the thermosensory input, which the worm could exploit, in principle, as an error signal to correct isothermal alignment. The simplest hypothesis is that the worm maintains isothermal alignment by responding to any temporal change in the thermosensory input driven by its own movement by countering that movement, an action that may be achieved by increasing the curvature of the trajectory of the head.
Mathematical modeling of sensorimotor control in isothermal tracking
Since C. elegans crawls by undulating with negligible lateral
slipping, it is possible to approximate its forward movement with mathematical
conciseness, by using a single-scalar quantity to describe the time-varying
curvature of the worm's head, which determines the subsequent path of the
body. Upon this framework, we build a minimal phenomenological model for
isothermal tracking as a direct transformation between thermosensory input and
the curvature of forward undulatory movement. We note that the simplifying
assumption of negligible lateral slip does not apply to all types of agar
surfaces. In many cases the slipping of the body of C. elegans slows
its forward speed by up to 20% relative to a no-slip condition
(Karbowski et al., 2006).
With the simplifying assumption of negligible lateral slip, the entire
undulating path of C. elegans sliding at speed
|v|, within each persistent period of forward movement,
is completely described by one variable 
(t), the angle of the
tangent to the worm's head with respect to a fixed axis in the plane, which we
define so that =
/2 when the head is pointing directly up the
spatial gradient. The undulating path can also be described in terms of the
time-varying curvature at the worm's head,
.
Therefore, a model of a crawling C. elegans during each isothermal
track only needs to take one mathematical conversion into account, between
changes in thermosensory input at the head (
)
and changes in curvature at the head (proportional to
).
Even in the absence of any temperature gradients,
=0, C. elegans crawls with
propulsive undulations with basal frequency 
and amplitude
0, so that we model its motion as a simple oscillator that
is determined by:
 | (1) |
where ()=d()/dt. Our minimal descriptive model of isothermal
tracking behavior contends that C. elegans simply curves more
vigorously to in response to any temporal change in the thermosensory input
(Fig. 6A). This strategy may be
incorporated by using a multiplicative factor in Eqn 1, allowing C.
elegans to change its curvature in proportion to changes in the
thermosensory input, :
 | (2) |
|
) is
dictated by considerations of symmetry, and minimally demands that a change in
the sign of (i.e. inverting the
temperature gradient) does not affect the response, i.e.
f()=f|()|.
In this discussion, we focus on the case
f()=g2,
where g is a scalar gain, for its analytical simplicity. A wide
variety of functional forms - e.g. those that are shown in
Fig. 6A, can produce
qualitatively similar results in numerical simulations.
In our experiments, the thermosensory input in a spatiotemporal thermal
gradients depends on (a) the amplitude and frequency of the superposed
temporal thermal gradient, (b) the speed and direction of the worm's head, and
(c) the steepness of the ambient spatial thermal gradient, so that:
 |
 | (3) |
where ()t=
()/t. At this point, the pair of
coupled non-linear differential equations, Eqn 2 and Eqn 3 and the functional
form of f(), provide a full
description of the dynamical system corresponding to the moving animal. All
parameters that we used in the model are summarized in
Table 1. We performed numerical
simulations of Eqn 2 and Eqn 3, fitting g to the experimental
observations of Fig. 5, and
found that we were able to reproduce the basic dynamics of real worms
navigating the spatiotemporal thermal gradients in the range of our
experiments (Fig. 7A-D).
|
|
Furthermore, several considerations allow us to develop an approximate
analytical solution to the model for sensorimotor control represented by Eqn 2
and Eqn 3. First, we are less interested in the detailed behavior of the
tangent to the worm's head (the parameter) than in the overall averaged
heading of the animal (
) in
response to spatiotemporal thermal gradients defined as:
 |
where 2/ is the period of undulation. Second, we know that
changes very little over single
undulation cycles, allowing us to analyze changes in
as first-order perturbations.
Third, in our experiments, we provide our temporal gradients with slow
frequencies (<<), so we are mainly interested in the limit in
which the superposed temporal thermal gradients can be viewed as constant in
time with respect to the faster oscillations driven by self-movement. We begin
by integrating Eqn 2 over the period of one undulation cycle
(T=2/), noting that the first term vanishes immediately:
 | (4) |
In order to integrate Eqn 4, we must incorporate Eqn 3 that describes
changes in the thermosensory input, ,
which is a function of the movement of the worm's head and of time. Since the
superposed temporal variations that characterize our experiments are slow with
respect to the undulation cycle and the fast variations in temperature are
driven by the movements of the worm's head in the spatial thermal gradient, we
may employ a perturbative expansion of
f() about the mean direction of the
animal's head, . This allows us
to calculate the first non-vanishing integrated term:
 | (5) |
Finally, for the case
f()=g2,
Eqn 5 yields the ordinary differential equation:
 | (6) |
for the averaged orientation variable
driven by a scaled temperature
gradient:
 |
that governs the relative contributions of the superposed temporal gradient
and the temporal gradient associated with locomotion. Here
is the scaled gain associated with the modulation of the curvature of the
head.
The fixed points of Eqn 6 have direct behavioral relevance. In the absence
of superposed temporal gradients
[
], the nonlinear oscillator
described by Eqn 6 has two stable fixed points at
which
correspond to the two stable isothermal tracks in different directions but
orthogonal to the spatial thermal gradient
(Fig. 6C). With shallow
superposed temporal gradients
) is
oscillatory, the worm will be driven in floral trajectories as the fixed
points oscillate over time. As the superposed temporal gradient approaches the
magnitude of the temporal variations driven by self-motion
(
) the two fixed-points
merge, so that 
is
allowed to slide through successive cycles of 2, and the worm may be
driven in looping trajectories. A convenient visualization of motile behavior
in spatiotemporal thermal gradients, within the context of our
phenomenological model, is a parametric plot of the fixed point
and the `driving
force'
(Fig. 6D,
Fig. 7E,F). In conclusion, the
phenomenology of C. elegans navigating a variety of spatiotemporal
thermal gradients can be qualitatively and quantitatively reconciled within
the parameter space of a nonlinear dynamical system that describes the
sensorimotor transformation for isothermal tracking.
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Discussion |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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).
However, the underlying principles of sensorimotor control that produce the
ability to track isotherms have not been examined. Here, we report that a
simple and direct sensorimotor link - between a sensor of thermosensory input
at the head and an effector that changes the curvature of the trajectory
beginning at the head - explains not only the ability of C. elegans
to maintain isothermal alignment in fixed spatial thermal gradients, but also
the more complex trajectories of C. elegans navigating spatiotemporal
thermal gradients. We found that the mathematical conciseness of describing forward crawling in C. elegans, which may be approximated using the time-varying curvature of the animal's head propelled at constant speed, admits the development of a minimal phenomenological model of the sensorimotor transformation for isothermal tracking. We had only to augment the description of forward crawling with a behavioral rule by which C. elegans changes the curvature of its forward undulatory movement in response to changes in thermosensory input. We found that the resulting model provides a compact and accurate description of isothermal tracking behavior that is consistent with behavior across a wide range of spatiotemporal thermal gradients.
A complete understanding of the neurobiology of C. elegans
isothermal tracking will require bridging our descriptive model of behavior
with mechanistic models of nervous system operation. To date, the
contributions of specific neurons to C. elegans thermotactic behavior
have largely been described in qualitative terms - ablating individual
neurons, such as the AFD sensory neurons or the AIY and AIZ interneurons, may
abolish isothermal tracking behavior at temperatures near its previous
cultivation tempeature or abolish cryophilic movement at temperatures above
its previous cultivation temperature, suggesting their general contribution to
these behaviors (Mori and Ohshima,
1995; Chung et al.,
2006). Progress will require defining the computational properties
of the nervous system during the processing of behaviorally relevant
thermosensory input. Once the physiological properties of the neural circuit
for isothermal tracking are defined, it should be possible to build
mechanistic neural network models that encode the basic operations of
isothermal tracking behavior. Our descriptive model, as it fits the isothermal
tracking behavior across a wide range of spatiotemporal thermal gradients, may
provide a guide for developing such mechanistic models.
The sensorimotor strategy that gives rise to isothermal tracking deserves
comparison with the sensorimotor strategy that enables C. elegans to
exhibit cryophilic movement. One difference is that the two different
behaviors are exhibited in different temperature ranges: the worm executes the
strategy for isothermal tracking in a temperature range within 2°C of the
previous cultivation temperature, and that for cryophilic movement in a
temperature range above the previous cultivation temperature
(Hedgecock and Russell, 1975;
Ryu and Samuel, 2002). Another
difference is that cryophilic movement involves a stochastic strategy and that
isothermal tracking involves deterministic maintenance of isothermal
alignment. However, an interesting similarity is that both strategies utilize
temporal processing of thermosensory input to assess the orientation of the
animal's movements with respect to the ambient spatial thermal gradient by
exploiting those same movements. During cryophilic movement, C.
elegans extends (shortens) periods of forward movement in response to
decreasing (increasing) temperature, and thus effects a biased random walk
towards colder temperatures (Ryu and
Samuel, 2002). In the case of isothermal tracking, it is the
side-to-side movements of the head in a spatial thermal gradient that drives
the temporal variations in temperature that allow C. elegans to
deterministically correct its isothermal alignment. Therefore, the
sensorimotor transformations underlying cryophilic movement and for isothermal
tracking have computational consequences, converting the scalar quantity of
thermosensory input into the vector quantity of motility, either average
velocity down thermal gradients during cryophilic movement or isothermal
alignment during isothermal tracking.
Finally, our observations allow us to estimate the thermal sensitivity of
C. elegans during isothermal tracking. If C. elegans tracks
isotherms by detecting temperature changes coupled to its own side-to-side
movements on gradients as shallow as 0.1°C cm-1, then it must
detect temperature changes as small as 0.005°C. C. elegans would
then approach the champion thermosensory metazoan, a blind cave beetle that
detects temperature changes as small as 0.001°C
(Corbiere-Tichane and Loftus,
1983).
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Acknowledgments |
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Footnotes |
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* These authors contributed equally to this work 
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References |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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=80 s; P>0.5). (E) Mean durations of isothermal
tracks of wild-type worms navigating radial spatial thermal gradients with
0.2, 0.7, and 1.2°C cm-1 steepness (black circles). Each point
represents data from 
approaches 1; the dotted line
shows 
. At point x,
|
as a function of 