Food transport in the C. elegans pharynx
Department of Molecular Biology, University of Texas Southwestern Medical Center, 5323 Harry Hines Boulevard, Dallas, Texas 75390-9148, USA
* Author for correspondence (e-mail: leon{at}eatworms.swmed.edu)
Accepted 3 April 2003
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Summary |
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Key words: nematode, Caenorhabditis elegans, pharynx, food transport, hydrodynamic simulation, feeding
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Introduction |
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C. elegans is, therefore, a filter feeder. It takes in liquid with suspended food particles, traps the particles, and expels the liquid. However, unlike many filter feeders, which separate food from medium by passing the suspension through a mesh that traps the food particles, there is no obvious filter in the pharynx through which particles and liquid are separated. A striking example of differential motion of particles and fluid is posteriorward transport of particles within the anterior half of the pharynx, the corpus (Fig. 1). During active feeding, bacteria are found at various points along the length of the relaxed corpus. Corpus muscles contract when a feeding motion (a pump) begins, and fluid rushes in at the mouth, sweeping food particles posteriorly. The contraction is followed by relaxation, which closes the lumen and expels the fluid. Unlike the fluid, the bacteria do not return to their original positions; when the motion is finished, they are seen to be located more posteriorly than they were before it started. This posteriorward transport is of course essential for the function of the pharynx, moving food from the mouth to the intestine.
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In our attempts to understand feeding C. elegans, we have
identified many defects, either genetic mutations or laser killing of
pharyngeal neurons, that result in inefficient feeding. Most of these fall
into two categories. In the first category are all those defects that decrease
the rate of feeding. Most or all of these defects affect the
excitatory motor neuron MC (Raizen et al.,
1995). It is obvious why worms that fail to excite pharyngeal
muscle pump slowly, and why slow pumping causes inefficient feeding. The
second category contains defects that affect transport of food in the
corpus. In laser killing experiments, the inhibitory motor neuron M3, which
controls the timing of pharyngeal muscle relaxation, is often a key component
of these defects (Avery,
1993b
). Genetically, transport defects result either from defects
in M3 function (Dent et al.,
1997
; Lee et al.,
1999
) or G protein signaling (see for example,
Robatzek et al., 2001
), which
probably also affects relaxation timing
(Niacaris and Avery, 2002
). It
is not obvious how these defects in relaxation timing cause poor transport. To
understand the genetics of feeding, it has therefore become necessary to
understand how the pharynx transports food.
Transport of bacteria within the corpus is puzzling, because the motions of
the corpus muscles during relaxation appear to be the reverse of the motions
during contraction. One might imagine that particles could be retained by
inertia during relaxation; heavy particles, slow to start moving, would lag
behind the liquid. However, in a system as small as the C. elegans
pharynx, fluid motions should be dominated by viscous (frictional) forces, and
inertial forces should be negligible. In systems of such low Reynolds number
(the ratio of inertial to viscous forces), motions are reversible and linearly
related to force (Vogel,
1994). Consequently, a reciprocal motion has no net effect
(Purcell, 1977
). The pharynx
should not work.
In this paper we show by analysis of videotapes that the motions of the anterior pharyngeal muscles during relaxation are not precisely the reverse of those during contraction. Corpus motions may be purely reciprocal, but the motions of the isthmus muscles immediately posterior to the corpus are slightly delayed, so that the isthmus is relaxed when the corpus begins to contract but contracted when the corpus relaxes. Simulation shows that this small deviation from perfectly reciprocal motion is sufficient to produce a small degree of net posteriorward particle transport if the particles are assumed to move at the same average speed as the fluid. However, transport is much less efficient in this simulation than in the animal. To resolve this discrepancy, we propose that the geometry of the pharyngeal lumen causes particles to be pushed to the center of the pharyngeal lumen during relaxation. Because fluid velocity is fastest at the center of a tube, the bacteria move faster than the mean fluid velocity. When this assumption is incorporated into the simulation, transport becomes efficient. The transport mechanism can be understood in simple, intuitive terms. In addition, this model explains why the genetic and microsurgical perturbations described above result in inefficient transport, and makes a prediction, which we confirm, that small bacteria will be better food than large bacteria.
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Materials and methods |
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Pharyngeal dimensions
To measure the outer dimensions of the pharynx, worms were mounted on pads
of 4% agar in M9 buffer (Sulston and
Hodgkin, 1988) containing 50 mmol l-1 sodium azide to
immobilize them, and measurements were taken from animals that were lying
straight or nearly straight. Six gravid adult hermaphrodites and ten
first-stage larvae (L1) mounting within 1 h and 11 min of hatching were
measured with an ocular micrometer, calibrated with a stage micrometer. The
results are shown in Table
1.
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To estimate lumen dimensions, we measured electron micrographs published by
Albertson and Thomson (1976).
For instance, fig. 5 of that
paper is a cross-section of the procorpus. The diameter of the procorpus in
that figure is 14.8 cm. Since the actual mean diameter of the adult procorpus
is 17.1 µm (Table 1B), the
printed electron micrograph is magnified 8600 times. The mean distance from
the center of the pharynx to the apices of the three radii of the relaxed
pharynx in the figure is 3.7 cm, which corresponds to 4.2 µm in the adult
procorpus. Assuming that the cuticle lining the pharyngeal lumen does not
shrink or stretch, the perimeter of the lumen is six times this, or 25 µm.
Finally, if the lumen cross-section is assumed to have the shape of an
equilateral triangle when fully open, its maximum diameter (defined as the
diameter of an inscribed circle) is the perimeter divided by 3
3, or 4.9
µm. Similar measurements of fig.
6 of Albertson and Thomson
(1976
), a section through the
isthmus, and application of the same calculations to the adult isthmus and to
the L1 isthmus and procorpus (assuming that the ratio of lumen diameter to
pharynx outer diameter is the same in L1s as in adults) give rise to the
estimates in Table 1C. While
there are several unverified assumptions in these calculations, they are
likely to be more accurate than can be made directly from video recordings of
the contracted pharynx, and they are broadly consistent with other
observations. For instance, 0.8 µm latex beads fit in the isthmus with
little room to spare, consistent with the calculated 1.0 µm diameter, and
adults can swallow 4-5 µm iron particles
(Avery and Horvitz, 1990
), but
only with difficulty, consistent with the estimated 3.9 µm diameter of the
isthmus, assuming that pressure from the corpus can force the isthmus lumen to
open slightly beyond a triangular cross-section. (If the isthmus opened to a
circular cross-section, a particle up to 6.4 µm in diameter could be
accommodated without cuticle stretching.)
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To estimate how lumen diameter varied along the length of the pharynx, an image of the pharynx was imported into a vector drawing program (Deneba Canvas) and a curve was drawn on it representing the approximate shape of the open lumen remembered from video recordings. (We could not draw direct from video images because the pharynx is never straight, in focus, and immobile enough to image sharply along the entire length of the anterior isthmus and corpus.) The procorpus and isthmus were drawn as constant in diameter along their lengths, and the metacorpus was represented by an elliptical section. The drawn shape was converted into a bitmap and diameters read from the resulting file. Finally, the dimensions were adjusted to match the measurements and estimates in Table 1A and C. In the simulations an L1 size pharynx transports 0.7 µm diameter particles (estimated size of a typical E. coli cell, based on comparison with 0.8 µm latex beads).
Isolation, identification, and photography of soil bacteria
Soil samples collected in five locations in the Dallas metroplex area were
suspended in water and plated at different dilutions on LB plates until single
bacterial colonies could be identified. Colonies that appeared homogeneous,
fast growing and different from each other were replated 2-3 times until clean
cultures were obtained. Identification of bacteria was done by commercial 16S
rDNA sequencing (Midilabs, 125 Sandy Drive Newark, DE 19713, USA;
www.midilabs.com),
and sequences were aligned against the provider's MicroSeq database. According
to the provider's recommendations, strains with less than 1% sequence
differences from the best database match were considered a species match,
strains with 1-3% difference from the closest match were considered a genus
match, isolates with >3% difference were considered not to match. For
photography, we removed bacteria from an NGM or NGMSR plate (identical to the
plates on which growth rate measurements were made) with a platinum wire and
transferred them to pads of 3% agarose in M9 salts, added a small amount of a
suspension of 0.8 µm blue-dyed latex beads (Sigma stock number L1398) and a
coverslip, and photographed them on Kodak elite chrome 200 slide film under
differential interference contrast optics.
Growth rate measurements
Growth rate was defined as an inverse of the number of days taken for an
animal to grow from the L1 stage to adulthood. Plates for growth rate
measurements were prepared as follows. Bacteria were grown for 1 or 2 days on
60 mm NGM plates to obtain a bacterial lawn covering approximately 50% of the
plate area. For most of our work, worms were grown on lawns of E.
coli HB101 or DA837 on NGMSR medium, which contains streptomycin and
nystatin to reduce growth of contaminants
(Davis et al., 1995). We also
used NGMSR for the measurement of growth rates on HB101 and DA837. Growth rate
measurements on the soil bacteria isolates were done instead on NGM
(Sulston and Hodgkin, 1988
),
since the soil bacteria isolates were sensitive to streptomycin. 50-100
starved synchronized L1s were transferred to a plate, which thereafter was
kept at 18°C and observed every 1-3 h until approximately 50% of the
animals reached the adult stage. The inverse of time (in days) it took
hermaphrodites to become adults was plotted as growth rate. Adults were
recognized by the appearance of a mature vulva, and by the production after
several hours of eggs. The error bars extend from the inverse of the time of
the first observation after that at which worms reached the adult stage to the
inverse of the time of the last observation before that at which worms reached
the adult stage. Worms were synchronized by egg starvation
(Emmons et al., 1979
): virgin
adult hermaphrodites were lysed in 40% bleach, 0.5 mol l-1 sodium
hydroxide for 5 min followed by three washes in M9, then eggs were incubated
overnight in M9 at 18°C to allow L1s to hatch.
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Results |
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This paper is concerned with transport of food by the corpus and anterior
isthmus during pumping. Transport is one of two functions of the corpus and
anterior isthmus, the other being trapping. The muscles of the corpus and
isthmus are radially oriented, so that when they contract, the lumen opens.
When the lumen of the corpus and anterior isthmus opens, it is filled by
liquid sucked in through the mouth. Suspended bacteria are carried in with the
liquid. Relaxation closes the lumen of the pharynx, expelling the liquid. Some
or all of the bacteria, however, remain in the pharynx. This is trapping. At
the beginning of the next pump, the previously trapped bacteria are found in
the lumen of the pharynx. When the muscles contract, the bacteria are carried
posteriorly by the inflow of liquid. They do not return to their original
position when the muscle relaxes, however; after relaxation they are found at
a more posterior position than they occupied before contraction. With repeated
pumps all food is carried back to the isthmus
(Avery, 1993b). This is
transport. Unfortunately, it is difficult to see by simple observation how
transport works. The relaxation is very rapid: the corpus usually goes from
fully open to closed in less than 1/60 of a second, the smallest time interval
that can be resolved in a standard NTSC video. It has therefore not been
possible to watch particle motion during relaxation.
Motions of beads during pumping
In an attempt to record the motions of particles during pumping, we fed
worms blue-dyed 0.8 µm latex beads, hoping that since they are
high-contrast objects under differential interference contrast, dyed beads
would be easier to see during rapid motion than bacteria. This experiment was
mostly unsuccessful; however, one useful sequence, shown in
Fig. 2, was found in several
minutes of recording. The sequence began with the corpus fully relaxed and 3
or 4 beads at its anterior end (Fig.
2A). 17 ms later the corpus had begun to contract, but the
particles had not moved. During the next 83
ms(Fig. 2C-G) the contraction
of the corpus continued, and the beads moved from the anterior end of the
corpus to the lumen of the metacorpus. In the next field
(Fig. 2H) the relaxation was in
progress. One bead had moved forward to the anterior metacorpus, and three to
the posterior procorpus. In the final field
(Fig. 2I) the relaxation was
complete. The beads did not move in the final 17 ms.
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This sequence demonstrates net posteriorward particle transport. The beads began in the anterior corpus and finished in the posterior corpus, although they moved both posteriorly and anteriorly during the pump. It also shows that particles do not always move with fluid in the pharyngeal lumen. This is most obvious in Fig. 2H,I. During this interval the corpus went from partially contracted to fully relaxed, and the liquid filling the metacorpus must have flowed anteriorly through the procorpus to the mouth. However, the four latex beads in the posterior procorpus and anterior metacorpus did not move.
Anterior isthmus motions are delayed
At low temporal resolution, the corpus and anterior isthmus appear to
undergo a simple reciprocal motion during pumping, in which the relaxation is
just the reverse of the contraction. As argued above, such a reciprocal motion
cannot easily explain net transport of food particles. We therefore examined
the motions of the corpus and isthmus using videotapes.
Within the resolution of this analysis, motions of the corpus were simultaneous along its entire length. However, as demonstrated in Fig. 3, the motions of the anterior isthmus were slightly delayed relative to the corpus. In this series, the first visible contraction of the corpus was seen at 33 ms. The isthmus was still closed at 100 ms; its first visible opening occurred in the next field, at 117 ms. Similarly, while the corpus went from fully contracted to relaxed between 150 and 167 ms, the anterior isthmus did not relax until 233 ms. Furthermore, the anterior isthmus did not contract simultaneously along its length. At 117 ms, only the anteriormost part of the isthmus was open. In the following frames, the opening extended progressively more posteriorly, as can be seen in the fields at 150, 167 and 183 ms. Thus, unlike the corpus, which contracted and relaxed as a unit, the anterior isthmus contracted in a wave that swept rapidly from anterior to posterior.
The precise timing of pharyngeal muscle motions is variable. Table 2 shows the delay of isthmus motions with respect to corpus motions in two animals. Neither of these is the animal of Fig. 3. The duration of the corpus contraction typically varies between 100 ms and 1 s. However, the features described above and demonstrated in Fig. 3 were consistent: corpus motions were simultaneous, but anterior isthmus motions were delayed with respect to the corpus, and both the opening and the closing of the anterior half of the isthmus began at the anterior end and progressed posteriorly to the middle. Because of the anterior isthmus motions, pharyngeal muscle motions during corpus relaxation were not the reverse of motions during contraction. For instance, the isthmus was closed during most of the corpus contraction, but was contracting while the corpus relaxed.
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Simulation of particle transport
To find out whether the delayed motions of the isthmus could account for
transport of particles in the corpus, we simulated the motions of fluid and
particles in the pharyngeal lumen. Particles were assumed to move according to
two rules. (1) When the diameter of the pharyngeal lumen (defined as the
diameter of an inscribed circle) is the same as or smaller than the particle
diameter, the particle is held by the walls of the lumen and unable to move
(Fig. 4). (2) When the diameter
of the pharyngeal lumen is greater than the particle diameter, particles are
free to move, and move with the fluid.
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Interaction between particles was not modeled, so the simulation is strictly valid only for a nearly empty pharynx.
Anterior isthmus delay produces inefficient transport
The pharyngeal muscles were assumed to move as summarized in
Fig. 5; timing was based on the
series of images from which Fig.
3 was abstracted. The corpus begins contracting at time 0, reaches
full contraction at 133 ms, and requires 17 ms to relax. Between 0 and 133 ms
and between 133 and 150 ms motions are linear, for simplicity. The anterior
end of the isthmus contracts from 83 to 150 ms and relaxes from 150 to 167 ms;
the middle of the isthmus contracts from 150 to 183 ms and relaxes from 183 to
200 ms. Between the anterior and middle isthmus, times of contraction and
relaxation vary linearly with position. The posterior half of the isthmus
remains closed through the pump. It was not included in the simulation, so
transported particles accumulate in mid-isthmus in the simulation. In reality
particles that reach mid-isthmus would eventually be carried back to the
terminal bulb by a subsequent posterior isthmus peristalsis.
Fig. 6A-D shows the simulated motion of three particles placed at the mouth, the middle of the procorpus, and the middle of the metacorpus. For comparison, Fig. 6E-H shows the movement of the same particles if the isthmus had contracted in synchrony with the corpus. As predicted, under these conditions there was no net particle movement; the particles returned to their starting positions (Fig. 6E-H). However, when the isthmus contraction was delayed, all particles finished slightly more posterior than they began (Fig. 6A-D). In fact, repetitive pumping would carry a particle from the mouth to the mid-isthmus. Fig. 7 shows the positions of a particle placed at the mouth after 1, 2,..., 14 pumps. The 14th pump carried it into the isthmus. From here it would be carried to the middle of the isthmus by subsequent pumps; in a real pharynx it would eventually be swallowed.
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Simulation reveals the existence of a point of no return. This is most clearly seen by looking at the most posterior particle (the red one) during corpus relaxation (Fig. 6C,D), which does not move. It is located at a stasis point where there is no flow. The stasis point is a watershed. All the liquid in the corpus lumen must leave it during relaxation, but there are two places it can go. Liquid anterior to the stasis point flows out of the mouth, and liquid posterior to it flows into the isthmus lumen, which is expanding at this time. While it lasts, the stasis point is a boundary that nothing can cross. The stasis point exists only during corpus relaxation; during contraction, flow is uniformly posterior. Thus a particle that moves posteriorly past this point during contraction can never cross it again. Any particle that reaches the point of no return will inevitably be carried into the isthmus and eventually swallowed.
The stasis point can also be thought of as the fulcrum of a lever. A particle at the stasis point remains stationary. A particle slightly posterior or anterior to the stasis point moves away from it at a rate and to a final distance that is roughly proportional to the particle's initial distance from it, just as the movement of a lever is proportional to distance from the fulcrum. This leverage explains how net transport is effected. During contraction, particles in the corpus are drawn back toward the middle of the isthmus, but during relaxation, they are pushed away from the stasis point. Because a particle in the corpus is always closer to the stasis point than to the middle of the isthmus, its anterior motion during relaxation is less than its posterior motion during contraction.
Centered particles are transported efficiently
Particles within the relaxed pharynx are consistently located on the center
line of the pharyngeal lumen (see, e.g.
Fig. 2). This can be explained
by the triradiate shape of the pharyngeal lumen, which would squeeze
off-center particles to the center as the lumen closes
(Fig. 8). A symmetrical
contraction of the pharyngeal muscles would be expected to release the
particle in the center of the lumen when it opens. Centered particles should
move differently from randomly located particles. A randomly located particle
will on the average move at the mean fluid velocity, as assumed in the
simulation above. However, fluid in the center of a tube moves faster than
fluid at the edges because the latter is slowed by friction with the walls. In
a tube with the triradiate shape of the pharyngeal lumen, fluid at the center
moves from 2.2 times faster than the mean flow velocity for a fully open lumen
to 3.5 times when the lumen is nearly closed (see Appendix). As this
acceleration will occur for both anterior and posteriorward movement, it is
not immediately obvious what effect it will have on net particle
transport.
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We repeated the simulation, but instead of assuming that particles moved at the mean flow velocity, allowed them to move at the center flow velocity. In addition, to roughly simulate the effect of a partially effective centering mechanism, we simulated the motion of particles moving at a constant factor times the mean flow velocity, with the constant ranging from 1.0 (i.e. mean flow velocity) to 2.0 (slightly less than the slowest expected for a perfectly centered particle). Fig. 9 shows the location of single particles placed at the mouth at time zero after a single contraction-relaxation cycle.
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Centered particles are transported far more effectively than particles moving at mean flow velocity. A centered particle is transported the entire length of the procorpus in a single pump (Fig. 9, blue particle). Only two cycles are needed to transport it to the isthmus, compared to 14 for a particle moving at mean fluid velocity. Even small accelerations produce an appreciable effect, and net transport increases more than linearly with velocity. A particle moving at 1.4 times mean flow velocity (Fig. 9, green particle) is transported almost twice as far in the first pump as one moving at mean flow velocity (Fig. 9, brown particle), and enters the isthmus after 8 pumps. A particle moving at twice mean flow velocity (Fig. 9, red particle) is transported more than half the length of the procorpus in the first pump, and enters the isthmus on the fourth.
Net transport of a centered particle, like transport of a particle moving at mean flow velocity, is absolutely dependent on delayed isthmus motions. If the corpus and isthmus contract together (as in Fig. 6E-H) a centered particle is pulled deep into the isthmus during contraction. But because the acceleration operates during anteriorward as well as posteriorward motion, it is transported back to its starting position during relaxation. Thus, the acceleration that results from centering particles does not in itself effect net transport. It merely magnifies the transport produced by delayed isthmus motion.
A simplified model: the four-stage pharynx
The essential features of transport within the corpus can be understood
using a simpler model shown in Fig.
10: the four-stage pharynx. In this model the isthmus and corpus
are each uniform in diameter along their entire lengths, and each contracts
and relaxes as a unit. Furthermore, their motions do not overlap: the corpus
contracts (Fig. 10A-C), then
the isthmus contracts (Fig.
10C,D), then the corpus relaxes
(Fig. 10D-F), then the isthmus
relaxes (Fig. 10F,G). Consider
the motion of a particle located at the mouth when corpus contraction begins.
At first, until the corpus diameter reaches the particle diameter, the
particle will remain at the mouth, held by the walls of the corpus lumen. When
the corpus opens to the particle diameter
(Fig. 10B), the particle will
be free to move, and will be sucked backward into the corpus
(Fig. 10C). Next, the isthmus
contracts, and more liquid is sucked in at the mouth and flows through the
corpus to fill the isthmus (Fig.
10D). The particle will move backward a small amount. In the third
stroke, the corpus relaxes. Liquid flows anteriorly, and at first, for as long
as the diameter of the corpus lumen exceeds the particle diameter, the
particle moves with the fluid. But when the lumen diameter equals the particle
diameter (Fig. 10E), the
particle is trapped within the corpus. Because of the triradiate shape of the
pharyngeal lumen (Fig. 4),
liquid can still flow around the particle. The particle remains where it is
while the corpus finishes its relaxation
(Fig. 10F) and the isthmus
relaxes (Fig. 10G). Like the
more realistic simulation, the four-stage pharynx has a point of no return:
the boundary between isthmus and corpus. A particle that moves posterior to
this point during contraction can never cross it again. (The four-stage model
does not correctly reproduce transport within the isthmus: subsequent pumps
will produce no net movement of a particle posterior to the corpus/isthmus
boundary.)
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The four-stage pharynx can be solved exactly. For instance, a particle
whose velocity is a times the mean fluid velocity moves distance
x from the mouth in a single contraction-relaxation cycle:
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As in the more realistic simulation, particle velocity has a more than linear effect. For instance, doubling particle velocity (from a=1 to a=2) increases sixfold the distance that a particle of 1/3 corpus diameter is transported. This nonlinear effect results because a fast-moving particle approaches the point of no return between the corpus and the isthmus more closely during contraction than a slow-moving particle does. During corpus contraction, a particle moving at twice the mean flow velocity moves to a point more posterior than one moving at mean flow velocity (Fig. 10A-C). Isthmus contraction pulls it further posterior, double the distance that a particle moving at mean flow velocity moves in the same step (Fig. 10C,D). However, during corpus relaxation, the particle is pushed forward only by the relaxation between it and the point of no return (Fig. 10D-F). A faster-moving particle, because it comes closer to the point of no return during contraction, moves less during relaxation.
Tests of simulation
Effect of relaxation timing
The timing of muscle relaxation seems to be critical for efficient
transport. The M3s are a pair of inhibitory motor neurons that innervate
corpus muscles. When they fire, they cause inhibitory postsynaptic potentials
in pharyngeal muscle, which can end the action potential, thereby causing
muscle relaxation (Avery,
1993b; Dent et al.,
1997
). When the M3s are killed along with all pharyngeal neurons
except the two main excitatory types MC and M4, relaxation is delayed and
bacteria are inefficiently transported. However, if all pharyngeal neurons
except MC, M4 and M3 are killed, relaxation is not delayed and transport is
efficient (Avery, 1993b
;
Avery and Horvitz, 1989
). These
results suggest that delayed relaxation of pharyngeal muscle results in
inefficient transport. We have described mutant strains in which bacterial
transport is inefficient (Avery,
1993a
). Several of these mutations affect G-protein signaling, and
may also affect the action or effectiveness of the M3s
(Niacaris and Avery, 2002
;
Robatzek et al., 2001
).
We suspected that these manipulations caused poor transport because they affected the relative timing of corpus and isthmus relaxation. To test whether the model could explain the importance of relaxation timing in this way, we simulated transport, holding isthmus motions constant but varying the timing of corpus relaxation, and measured the distance a centered particle placed at the mouth moved. Fig. 11 shows that corpus relaxation time is indeed critical, with transport maximal in a narrow window from 133 to 167 ms. Furthermore, the normal timing measured from video sequences produces near maximal transport.
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Effect of particle size
Both the four-stage model and the simulation predict that particles that
are small compared to the corpus lumen diameter will be transported more
efficiently. Several observations suggest that the size of the pharynx is
indeed important for effective feeding. The time immediately after hatching,
when the pharynx is at its smallest, is critical for many feeding-impaired
mutants. These mutants often fail to grow after hatching and may arrest for
days, but once they pass the first larval stage they grow rapidly and become
fairly normal adults. This L1 arrest is probably caused by a failure to
ingest, because it can be relieved by changing the food (data not shown).
Although the pharynx is at its smallest in absolute terms in the newly hatched
larva, it is large compared to the worm itself, occupying 29.6±1.3% of
the length (71.6±1.0 µm for the pharynx, 242±10 µm for the
worm; see Table 1). The pharynx
grows rapidly during the first larval stage, but thereafter more slowly than
the rest of the animal (J. Hodgkin, personal communication), so that in the
adult hermaphrodite it occupies only 14.4±0.4% of the length
(144.7±0.9 µm/1002±28 µm;
Table 1). The increase in
diameter is greater than the increase in length. For example, the diameter of
the isthmus increases almost fourfold from 4.0±0.1 µm in the newly
hatched larva to 15.3±0.3 µm in the adult
(Table 1). The large relative
size of the pharynx at hatching and its early growth may be caused by
selection for attainment of efficient size as early as possible. These
observations all suggest that a small pharynx functions poorly compared to a
large one.
To test whether food particle size is important we isolated several bacteria from soil and measured how well wild-type and various feeding-defective mutants grew on them. Food is not limiting for wild-type growth on good food sources; feeding-defective mutants allow improved discrimination. Fig. 12 shows the results. Bacterial strains are shown in order of decreasing ability to support growth. As predicted, cell size increases as edibility decreases. The least edible bacteria, in addition to being large in size, form spores (refractile cells in photographs). However, we do not believe that spores are poor at supporting growth, because a spontaneous sporulation-defective mutant of Bacillus megaterium L10 is even worse at supporting growth than the parent strain. Furthermore, the correlation between size and edibility is better than between sporulation and edibility. For instance, Bacillus simplex, which makes small spores, supports growth about as well as Pantoea dispersa, a bacterium of about the same size that was not seen to form spores.
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There is one conspicuous exception to the inverse correlation between size and edibility: E. coli strain DA837. It is the same size as E. coli HB101 but a far worse food source. A lawn of DA837 on a plate of worm medium has a sticky, tacky texture that can be felt with a platinum pick. In contrast, lawns of the other bacterial strains in Fig. 12, including E. coli HB101, did not detectably resist the passage of a pick through the bacteria. When bacteria of these strains taken from a worm medium plate are suspended in liquid, they move as individual cells that do not adhere to each other. In contrast, DA837 bacteria are seen in clumps of several cells. (Unfortunately this difference is obvious only when bacteria are suspended free in liquid, and Brownian motion makes it difficult to photograph.) It may be that worms eating DA837 must deal with them as clumps of cells rather than single cells, and that the effective particle size is therefore larger. Alternatively, DA837 may be handled poorly by the pharynx at some step other than transport. This is suggested by the differential effect of worm genotype on ability to grow on DA837: eat-5 worms are particularly affected. The main effect of eat-5 is to slow down terminal bulb contractions, with little effect on the corpus.
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Discussion |
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An alternative transport mechanism could be based on posterior-moving waves
of contraction or relaxation along the length of the corpus. Indeed, in a
previous analysis of corpus and isthmus motions, Seymour et al.
(1983) reported that there are
such waves of motion: 'successive regions of the triradiate oesophageal
[i.e. pharyngeal] lumen are opened by their radial muscles, beginning with the
procorpus and followed by the metacorpus and isthmus' and 'As soon as
the bacteria reach the mid procorpus the whole oesophageal lumen begins to
close from the anterior end backward'. While such motions would easily
explain net particle transport, we have been unable to confirm that they
occur. Seymour et al.'s statement was based on 24 frame s-1
ciné recordings of unc-15 and unc-51 mutant animals,
since they were unable to get useful recordings from the more active wild-type
animals. We observe that both of these mutants have obviously abnormal feeding
behavior: for instance, they pump more slowly than wild type (L. Avery,
unpublished observation). Our video recordings, which could be analyzed at a
resolution of 1/60 s, were made from wild-type and unc-29 animals.
unc-29 encodes a body muscle-specific nicotinic acetylcholine
receptor subunit (Fleming et al.,
1997
) and has no effect on pharyngeal muscle motions. In these
recordings the corpus appears to open and to close simultaneously along its
entire length, and to have perfectly reciprocal motions. We have also looked
at a videotape made from Seymour et al.'s ciné recordings, and at a
high-speed video recording made by Lars Philipson (personal communication). In
none of these recordings could we see any hint that the anterior corpus opens
or closes before the posterior. Indeed, in Philipson's high-speed recording it
sometimes appeared that the closing of the procorpus began slightly
after that of the metacorpus, as if the pressure of the fluid being
forced out of the metacorpus briefly resisted the closing of the procorpus.
This slight delay, if it had an effect, would tend to result in anteriorward
particle transport. Thus, we do not believe that transport is effected by
waves of corpus muscle motion.
Simulation of particle transport
A pharynx in which motions are perfectly reciprocal and in which particles
when free move at mean fluid velocity would clearly not produce net transport.
It was intuitively clear that delayed isthmus motions, while they are a
deviation in the right direction from reciprocity, would produce at best
inefficient transport of particles moving at mean fluid velocity. Without
delayed isthmus contraction, there would obviously be no net transport, even
of particles that move faster than mean fluid velocity. But unassisted
intuition failed to predict how these two deviations from the reciprocal/mean
velocity case would interact: would faster particle motion enhance or weaken
the effect of the delayed isthmus contraction, and if so, by how much? To
answer this question we found it necessary to simulate fluid and particle
motions. Interestingly, particle motion faster than mean velocity powerfully
(more than linearly) enhanced the effect of delayed isthmus motions.
Furthermore, with the aid of the simulation, it was possible to understand how
the two factors interact and the synergism: delayed isthmus motions create a
stasis point during corpus relaxation, and accelerated particle motion allows
particles to approach more closely or pass this point of no return. With this
new intuitive understanding in hand, we developed a simpler model, the
four-stage pharynx which, while not geometrically or quantitatively accurate,
captures the essentials of transport in an easily understandable way.
In our model particle motion is determined by three assumptions. (1) When the diameter of the pharyngeal lumen is smaller than the particle diameter, the particle is unable to move. (2) When the diameter of the pharyngeal lumen is greater than the particle diameter, particles are free to move and move with the fluid. (3) Particles are located at the center of the pharynx and move at the center flow velocity.
Assumptions 1 and 2 are intuitively plausible and consistent with observation (e.g. Fig. 2). Assumption 3, the most important novelty of the proposed mechanism, is hypothetical and cannot easily be verified by observation. Indeed, it is certainly not exactly true; with time some displacement from the center is inevitable (and is indeed visible in Fig. 2G). However, several arguments suggest that faster motion at the center is likely to contribute to particle transport. First, bacteria are consistently located at the center of the relaxed pharynx, as shown for instance in Fig. 2I. As the pharynx begins to contract, a particle must remain near the center during the time period immediately after it is released, because it will simply not fit anywhere else. This period is the most important for transport, because the cross-sectional area of the lumen is small, meaning higher fluid velocity for a given flow rate. As the contraction proceeds, a particle may tend to leave the precise center of the lumen, but as long as it remains away from the walls it will move faster than the mean flow velocity, and our simulation shows that even a small acceleration improves transport (Fig. 9). Moreover, diffusion of a particle away from the center will not necessarily make transport worse: if the particle is approximately centered during contraction but off-center during relaxation it will move more rapidly posteriorly than it does anteriorly, resulting in even greater net transport.
Every application of mathematics to the real world requires approximations.
In this simulation the approximations are many, including the following. (1)
Interaction of particles was not modeled. (2) Free particles were assumed to
move at center fluid velocity. In reality, they would move slower because of
their finite size, and because they would tend to be pushed away from the
center by Brownian motion and less than perfectly symmetric muscle motions.
(3) The pharyngeal lumen was assumed to have a simple triradiate shape
(Fig. 4); the real
cross-section, however, is more complex and varies along the length of the
pharynx (Albertson and Thomson,
1976). (4) The diameter of the lumen as a function of position
along the pharynx is more complex in reality than in the model. (5) Muscle
motions were assumed to be piecewise linear functions of time
(Fig. 5). Assumption 1 was made
for computational convenience and because it represents an important limiting
case, but assumptions 2-5 were necessary because of lack of information, since
we do not have the data to replace these assumptions with more realistic
parameters. (Also, the shape and motions of the pharynx vary with age and
environment, leading us to suspect that some of these quantitative details
might not matter greatly.) For these reasons, we cannot claim that the
simulation models particle transport in a quantitatively accurate way
(although it may). We feel, however, that it probably captures the key
qualitative features of transport, for two reasons. First, as described above,
its workings are intuitively understandable in a way that does not depend on
the suspect quantitative details of motion or geometry. Second, it resembles
the real pharynx, both in the efficiency with which it transports particles,
and in the effect of certain perturbations on transport.
The power of the model is shown by its ability to explain a past observation and to make a new prediction. The past observation is that the timing of corpus relaxation is important for effective particle transport. Intuitively our model explains this because corpus motions must occur before isthmus motions for transport to occur. Simulation shows in fact that timing does matter, and that the actual muscle motions measured from videotape are near-optimal (Fig. 11). The new prediction is that smaller particles will be transported more effectively, and therefore that there will be an inverse correlation between bacterial size and ability to sustain C elegans growth. This is not an intuitively obvious prediction: while it is clear that bacteria must fit in the pharynx to be transported, it is not obvious without understanding the mechanism why a large particle that fits in the lumen will be transported less well than a small one. Indeed, in more conventional filter-feeding systems, larger particles are more easily separated from fluid than small ones. However, it was borne out by measurements of growth on different soil bacteria isolates (Fig. 12).
Particle trapping versus particle transport
Although we have focused on particle transport, the model we propose is
also capable of trapping particles. With simulation parameters based on
experimental measurements, bacteria that enter the mouth within 58 ms of the
beginning of corpus contraction are trapped within the corpus. Bacteria that
enter after 58 ms are expelled during corpus relaxation. However, it is
virtually certain that the simulation does not trap particles in the same way
as the real pharynx. First, trapping seems to be more effective in the real
pharynx; particles that enter the pharynx are rarely seen to leave again.
Second, Seymour et al. (1983)
reported (and we have confirmed) that bacteria usually do not enter the corpus
until near the end of the contraction, precisely the time at which they would
be least effectively trapped in the simulation. Normally the metastomal flaps
at the anterior end of the pharynx block bacteria from entering during most of
the corpus contraction; instead, they pile up in the buccal cavity, a sort of
vestibule to the pharynx. Just before corpus relaxation the metastomal flaps
move, allowing the bacteria to enter the corpus lumen, where they are trapped
by a mechanism that remains mysterious. It seems likely that the metastomal
flaps and the small pm1 and pm2 muscles that operate them are somehow
involved, but because of their small size and the rapidity of the events, we
have been unable to see how they operate.
Conclusion
Our results do not establish the mechanism of particle transport in the
C. elegans pharynx. Direct confirmation would require high-speed
video recordings of bacteria-sized particles moving in the lumen of the
pharynx during pumping, which is technically difficult. We can say, however,
that no exotic mechanisms are required to explain particle transport. By
combining the observed muscle motions with three assumptions, two of them not
controversial and one plausible though unproven, we can reproduce in
simulation the efficient transport observed in the animal.
![]() |
Appendix |
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![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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O is the parameter used to describe pharyngeal motions, and is
plotted in Fig. 5. A and
O vary with time, but are invariant along the length of a
section. The volume of a section is
lA=lAmaxO. The four-stage pharynx
consists of just two sections: corpus and isthmus. In the more realistic
simulation, the entire procorpus was modeled as one section, since it was
assumed to be constant in diameter and to move as a unit. The isthmus,
although constant in diameter along most of its length, was modeled as a
series of small sections of length l57 nm, because the timing of
its motions varied along the length, and the metacorpus was also modeled as a
series of small sections since it varies in diameter. x, position
along the anterior-posterior axis, was set to 0 at the middle of the isthmus,
and increased in the anterior direction. This the posteriormost section,
section 1, extends from x=0 to x=l1, the
next most posterior section from x=l1 to
x=l1+l2, etc.
To simulate the movement of a particle of finite size, it was necessary to calculate the radius of the largest circle that can be inscribed in the lumen at a given O, so that we could determine when the particle was held by the walls and when free to move. This required that we choose a particular geometry for the pharyngeal lumen. We assumed that the lumen varied between a Y-shaped structure when fully closed to an equilateral triangle when fully open (Fig. 4). We also assumed that the apices of the lumen did not move. [This is unlikely; the apices probably move away from the center somewhat as the pharynx contracts. However, we did not have the data to model this more accurately, and in any case it has little effect on the simulation; it simply means that the true value of O at which a particle will be caught is slightly smaller than the one we used, i.e. our approximation is effectively the same as making the particle slightly bigger. This assumption also affects the calculation of center fluid velocity as a function of O (see below), but since the ratio of center to mean velocity varies only slowly with O, this should have little effect.]
rmax is the radius of a circle inscribed in the lumen
open to its maximum extent, when it is assumed to have an equilateral triangle
cross-section. Equivalently, rmax is the distance from the
center of the fully open pharyngeal lumen to any of the three closest points
on the lumen wall (Fig. 13).
![]() | (2) |
![]() | (3) |
|
Fluid and particle motions
Because we assumed the posterior isthmus to remain closed, the lumen had to
fill and empty through the mouth. Flow through the mouth is therefore
just the rate of change of the volume of the pharyngeal lumen. Flow through a
cross-section at position x is equal to the rate of change of the
volume of the lumen posterior to x. In particular, flow through the
anterior boundary of section n is equal to the sum of the rates of
change of volume of sections 1 through n:
![]() | (4) |
For x within a section extending from xp to
xa=xp+l, is linear
with x:
![]() | (5) |
![]() | (6) |
A particle moving at mean flow velocity follows the differential equation:
![]() | (7) |
![]() | (8) |
![]() | (9) |
An important special case occurs when (x) and
p are opposite in sign. For instance, consider a section into
which liquid is flowing at both the anterior and posterior ends. This requires
that the pharynx posterior to the section be relaxing and expelling liquid, so
that
p is negative, and that the section be contracting, so
that
>-
p, making
a=
+
p>0.
If we try to use Equation 9 to calculate when a particle that begins at
xp will reach xa, we get
t=-A0l/
a, a negative
number. But since the particle is moving in the anterior direction, toward
xa, this answer cannot be correct. In fact, the particle
will never reach xa. There is a watershed or stasis point
within the section at
x=xa-
a/(
a-
p)
where
(x)=0. Nothing can cross this boundary. It is such a
stasis point that leads to the point of no return in the posterior metacorpus
that is so important to understanding how the model transports bacteria. This
special case also occurs for accelerated particles and centered particles
(below).
Simulation method
Motions of particles, fluid and the pharyngeal lumen were simulated by a
discrete event method. Between events we used Equation 8 to calculate the
movement of particles moving at mean fluid velocity. An 'event' is any
discontinuity at which the equation becomes invalid or at which its parameters
change. The most important events are the release of a caught particle by the
contracting pharyngeal lumen, the catching of a free particle by the relaxing
lumen, motion breakpoints of the section the particle is currently in or any
posterior section, and the movement of the particle into an adjacent section.
The times of all these events can be predicted: the first three from particle
size and the predetermined pharyngeal motions, and the last from Equation 9.
The simulation was projected forward to the earliest of them, and Equation 8
was used to calculate particle position at the time of the event. Event times
were then recalculated and the simulation projected forward to the next event,
etc. Because positions were calculated from analytical solutions to the
differential equations governing motion, time is effectively continuous. That
is, we did not approximate time as a series of discrete steps during which
motions followed simplified approximations, as is necessary when differential
equations are solved numerically. Similarly, particles move in a continuous
space, although a discrete spatial approximation was introduced in modeling
the pharynx as a series of sections.
In addition to the four events at which motions change, we also defined 'snapshot' events: these were predetermined time points at which all particle positions and lumen diameters were calculated for use in drawing images like those shown in Figs 6 and 7. Yet another event type was used to place new particles at specified positions along the pharynx (most often at the mouth) at specified times. The simulation, written in the java programming language (http://java.sun.com), is available at http://eatworms.swmed.edu/~leon/pharynx_sim/.
Accelerated particles
The motions of a particle whose velocity is the mean fluid velocity
multiplied by a constant factor a are solved in a similar manner.
Equation 7 becomes:
![]() | (10) |
![]() | (11) |
![]() | (12) |
Centered particles
A particle that moves at the velocity of fluid in the center of the
pharyngeal lumen also follows Equation 10, but a is now a function of
O and therefore of time. We used a local Poiseuille flow
approximation to determine how flow velocity varied across the cross-section
of the lumen. This approximation is valid if inertial forces can be neglected,
and if flow is axial or nearly so (i.e. if lumen diameter changes slowly with
x). Velocity then follows Poisson's equation:
![]() | (13) |
![]() | (14) |
Equation 13 was solved numerically by simulated diffusion on a lattice. The
method is based on that described by Hunt et al.
(1995), but we made one
important change. Hunt et al.
(1995
) calculated the average
number of steps that a particle placed in the interior of the lumen takes to
reach the wall. We instead counted the average number of times a diffusing
particle hit each lattice point. This also solves the finite difference
version of Equation 13, but is computationally more efficient since it
requires only a simple increment of the hit count at each step. The C source
for our program is available at
http://eatworms.swmed.edu/Worm_labs/Avery/flowsim/.
The program takes input on the geometry (necessary to specify the boundary
condition Equation 14) in raster image files. Using a commercial drawing
package (Deneba Canvas), we created 10 images of the pharyngeal lumen for
O=0.1, 0.2,..., 1.0 on a 289x250 pixel lattice and solved
Poisson's equation for each. We show the solution for O=0.3 in
Fig. 14 as an example. At this
value of O, a, the ratio of center to mean flow velocity, was 2.95.
For the series, a varied from 2.21 at O=1.0. to 3.32 at
O=0.1, and was well fit (r2=0.9993) by the
quadratic a=1.0097O2-2.366O+3.5587. The
solution of Equation 10 is:
![]() | (15) |
![]() | (16) |
|
![]() |
Acknowledgments |
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References |
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