Ontogenetic Scaling of Burrowing Forces in the Earthworm Lumbricus Terrestris

Animal skeletons perform a number of important mechanical functions: they provide postural support, re-extend shortened muscles (enabling reversible shape changes) and transmit forces from muscles to other parts of the organism and to the environment (enabling feeding and locomotion; for reviews, see Chapman, 1950; Currey, 1970; Trueman, 1975; Wainwright, 1988). Since skeletons function throughout the lifetime of animals, the ontogenetic scaling of skeletal function may have important implications for the life history, ecology and evolution of these animals (e.g. Peters, 1983; Calder, 1984; Werner and Gilliam, 1984; Carrier, 1996). The purpose of the present paper was to examine the ontogeny of skeletal force transmission in the hydrostatic skeleton of a soft-bodied organism.

Several studies have examined ontogenetic scaling of skeletal function in organisms with rigid, lever-like skeletons, namely vertebrates and arthropods (e.g. Garland, 1985; Marsh, 1988; Queathem, 1991; Katz and Gosline, 1992; Carrier, 1995; Hughes et al., 1995). However, the hydrostatic skeletons of soft-bodied organisms function by a different mechanism from lever-like skeletons. Hydrostatic skeletons are generally composed of a deformable body wall in tension surrounding fluid and/or soft tissues in compression. Fluid in compression becomes pressurized; it is this pressure, rather than a rigid lever, that enables antagonism of muscles and stiffening of the organism (for reviews, see Chapman, 1958, 1975; Clark, 1964; Trueman, 1975; Kier and Smith, 1985; Wainwright, 1988). Since muscle forces are transmitted to other muscles and to the environment via the pressurized internal fluid, motions and forces in hydrostats are determined primarily by changes in the dimensions of the body segments themselves rather than changes in joint angles between segments. Diverse organisms from many phyla employ such a skeletal design. How does the ontogenetic increase in size of a hydrostatic skeleton affect its ability to transmit forces?

I chose earthworm burrowing as my experimental system to examine the ontogenetic scaling of force production in a hydrostat. Lumbricus terrestris is one of the largest of the temperate earthworms and has been studied often because of its abundance and agricultural importance (for reviews, see Darwin, 1881; Minnich, 1977; Edwards and Bohlen, 1996). L. terrestris grows by more than three orders of magnitude in body mass from hatchling (0.01 g; K. J. Quillin, personal observation) to maturity (5–10 g; Lakhani and Satchell, 1970), providing a wide size range for the study of body size. Furthermore, earthworms are excellent burrowers (see Darwin, 1881). Several researchers have measured burrowing force in adult earthworms using flat glass bridges (Manton, 1965; Seymour, 1970), artificial burrows (Keudel and Schrader, 1999) and soil discs (McKenzie and Dexter, 1988a,b). Others have calculated force from measurements of internal pressure (Chapman, 1950; Newell, 1950). However, none of these studies measured burrowing force across an ontogenetic size range.

The earthworm L. terrestris lives in semi-permanent burrows and feeds primarily by pulling leaves into the burrows and secreting enzymatic saliva to digest them externally (e.g. Darwin, 1881; Arthur, 1965). This species also eats soil, but generally only when food is scarce or when it cannot otherwise penetrate compacted soil (Darwin, 1881; Evans, 1947; Arthur, 1965). Thus, burrows are generally created by ‘crevice burrowing’ (Seymour, 1970) rather than by ingestion. The anteriormost segments are first elongated axially, inserted into crevices in the soil and then expanded radially to enlarge the diameter of the crevice. After a burrow has been established, it is often maintained (sometimes for weeks or months; Evans, 1947) by radial compaction of the walls.

Many soft-bodied crawlers and burrowers use peristalsis to translate local shape changes into locomotion (e.g. earthworms, sea cucumbers, caterpillars, snails; Alexander, 1983; Trueman, 1975; Trueman and Jones, 1977; Elder and Trueman, 1980). Peristalsis occurs when waves of alternating contraction and relaxation of muscles move along a radially flexible tube (Heffernan and Wainwright, 1974). Typically, the radially expanded regions anchor the organism to the substratum while the radially contracted, axially elongated regions advance over or through the substratum. When the longitudinal muscles of an earthworm segment contract, the segment becomes shorter (along the anterior–posterior axis) and wider and can exert radial forces. When the circumferential muscles of an earthworm segment contract, the segment becomes longer and thinner and can exert axial forces. Waves of circumferential and longitudinal muscle contractions pass posteriorly along the constant-volume body segments as a worm crawls forward, forming retrograde waves (waves that move in the opposite direction to locomotion; Gray and Lissmann, 1938; Newell, 1950; for a review, see Trueman, 1975). Peristalsis is used both for crawling on the surface and for burrowing within the soil.

The objective of the present study was to determine how burrowing forces scale with body size during earthworm ontogeny. The first experimental goal was to measure axial and radial forces exerted by burrowing earthworms and to do so using an apparatus that simulates an actual soil burrow. If muscle properties are constant as a function of body size, geometrically similar organisms are expected to exert forces that are proportional to body mass m_b^2/3, where m_b is body mass (e.g. Alexander et al., 1981; Alexander, 1985). This null hypothesis is derived from the observation that muscle force in vertebrates is proportional to the cross-sectional area of the muscle if peak muscle stress is constant (e.g. Close, 1972). Area scales as the square of length, or as m_b^2/3 in geometrically similar organisms (e.g. Thompson, 1917). The ontogenetic scaling of muscle properties is unknown for earthworms, but the scaling relationship force∝m_b^2/3 is used as the null hypothesis in the present study. Thus, the corollaries to this null hypothesis are that force divided by weight scales as m_b^−1/3 and force per area of application of force scales as m_b⁰.

Quillin (1998) observed geometric similarity in earthworms across an ontogenetic size range. However, muscle area was not one of the variables measured. The second experimental goal of the present study was to measure the scaling of muscle area since it is a critical determinant of the scaling of force production. Finally, Quillin (1999) observed kinematic similarity in earthworms that were crawling on a surface, but the kinematic scaling of burrowing earthworms has not been examined. Since force and velocity tend to be inversely related, the third experimental goal of this study was to measure the scaling of burrowing kinematics to provide a context for the observed scaling of burrowing forces.

I designed the force-measuring apparatus to meet two requirements: (i) the use of soil burrows similar to those the earthworms experience in nature, and (ii) the use of interchangeable parts to accommodate earthworms varying by an order of magnitude in length and diameter. I designed the apparatus to operate in two configurations. The ‘open’ burrow configuration enabled the measurement of radial forces and burrowing kinematics as an earthworm crawled through a cylindrical soil burrow (Fig. 1A,B). The ‘dead-end’ burrow configuration enabled the simultaneous measurement of radial and axial forces as an earthworm attempted to penetrate the end of a dead-end soil burrow (Fig. 1C,D). These two configurations mimic the circumstances that earthworms of the species Lumbricus terrestris experience in their semi-permanent burrows in nature. Seven sets of parts were built to accommodate seven size classes of earthworm ranging from 0.012 to 8.9 g.

The open-burrow configuration consisted of a short soil burrow (less than one body length) in series with a cylindrical force transducer and a long soil burrow (greater than one body length; Fig. 1A). I formed the burrows by packing soil around the outside of pipes and wires of known diameter (see below for soil properties). A Plexiglas lid served as the roof of the burrow and enabled observation of the earthworm within. The short length of the first burrow motivated the earthworms to crawl forward through the force transducer. The long length of the second burrow motivated the earthworms to crawl completely through the force transducer before pausing at the end of the burrow, a typical behavior.

Each radial transducer tip consisted of a metal or Plexiglas cylinder whose length and inside diameter were the same as the inside diameter of the burrow (as determined by the outside diameter of the pipe or wire that was used to form the burrow). Each cylinder consisted of top and bottom halves (Fig. 1B).

Each top half was attached to a pipe that fitted over the stiff metal rod at the end of the Plexiglas force beam. Each bottom half was attached to a base for placement between the two burrows. As an earthworm passed through the cylinder and contracted its longitudinal muscles (causing the segments within the cylinder to expand radially), a radial force was exerted against the top half of the cylinder, causing the force beam to be deflected. The two strain gauges on the force beam (half-Wheatstone bridge) transmitted the deflection in millivolts to LabVIEW software (Version 3.0.1; National Instruments, Austin, TX, USA) on a Gateway 2000 computer. Force data were collected at 10 scans s⁻¹ at 22 °C in daylight; scanning at a higher rate did not influence the magnitude of the force peaks. I calibrated the force beam with weights at the beginning of each set of experiments to derive a conversion from millivolt output to newtons. The calibration data were all linear (r²=0.999 or better). Plexiglas force beams are sometimes subject to creep and can, therefore, show significant hysteresis. I cannot eliminate the possibility that hysterisis affected the shape of the force peaks during the experiment. However, the transducer re-zeroed itself between force peaks and I only measured maximum force, not the shape of the force curve. Therefore, I do not believe that the material properties of the force beam caused error in the force measurements.

I measured peak radial force for six replicates for each of 36 individuals ranging in mass from 0.012 to 8.9 g. I began collecting force data the instant that the anteriormost segment of the earthworm entered the transducer cylinder and terminated data collection either when the last segment entered the cylinder or when the earthworm stopped crawling part-way through. Using this method, I obtained the duration of crawling from the duration of the force recording. By measuring the length of earthworm that had passed through the cylinder during the duration of the force recording, I was able to calculate average crawling speed through the force transducer. I also used the force recording to determine the number of cycles of peristalsis that occurred during the duration of the trial, which enabled me to calculate stride frequency. I then calculated average stride length as the product of crawling speed and stride frequency. I also measured the entire length of the earthworm as it crawled through the transducer so that the ratio of total crawling length to resting body length (and thus the degree of body elongation) could be calculated; resting body length was calculated from body mass using the scaling relationship described by Quillin (1998). After each replicate, I repacked the burrow. I also re-zeroed the force transducer using a micromanipulator to place the top half of the radial transducer cylinder over the bottom half such that the two were barely touching and not prestressed. I ran the six replicates for each individual consecutively; there were no consistent trends in the peak force values over the course of the six replicates.

The dead-end burrow configuration consisted of a short soil burrow in series with a cylindrical radial transducer and a conical axial transducer (Fig. 1C). The burrow and radial transducer were assembled as described for the open-burrow configuration. Each axial transducer cone consisted of a clear Plexiglas block drilled with a conical hole (apex angle 37 °) whose base diameter equaled the inside diameter of the burrow and radial transducer (Fig. 1D). The clear Plexiglas block enabled observation of the earthworm within the cone. Each cone was attached to a pipe that fitted over the stiff metal rod at the end of the Plexiglas force beam. When an earthworm inserted its anteriormost segments into the cone and pushed forward in an attempt to advance, the force beam was deflected and its output was recorded by LabVIEW software. I measured peak radial force and peak axial force for four replicates for each of 37 individuals ranging in size from 0.012 to 8.9 g. I did not run the four replicates for each individual consecutively; three or four individuals were measured during each session, alternating individuals for each run to give the earthworms a rest between replicates (without a pause, earthworms often ‘remembered’ that the burrow was impenetrable and would not make a second attempt). I observed no trend in the peak force over the course of the four replicates. The axial force beam was calibrated as described for the radial force beam. The burrows and transducer tips were reset between replicates.

The objective of this control experiment was to choose a ratio of earthworm diameter to burrow diameter for the scaling experiment that would maximize burrowing force while enabling replicate measurements. Peak radial force was measured as a 6.1 g earthworm (9.5 mm maximum resting diameter) crawled through the open-burrow apparatus configured with four different burrow diameters: 5.4 mm, 7.1 mm, 8.7 mm and infinite (a surface), which translated to burrow diameter to earthworm diameter ratios of 0.57, 0.75, 0.92 and infinity. Five replicates were recorded for each burrow configuration, and the order of the configurations was mixed (both starting and ending with the smallest diameter to ensure that the earthworm did not fatigue). Data were collected at 40 scans s⁻¹ at 22 °C in daylight.

The earthworm exerted highest forces in the narrowest burrow (Kruskal–Wallis, P=0.01; Fig. 2A). The earthworm also exerted the greatest number of radial pushes per run in the narrowest burrow in an apparent attempt to enlarge the burrow (P<0.001; Fig. 2B). However, the earthworm crawled much more slowly in the narrow burrow (P<0.001; Fig. 2C), often backing out after a very short distance of penetration and/or refusing to re-enter the burrow altogether. The earthworm did not change the extent of elongation of its body to accommodate different burrow diameter, as might be expected (P>0.10; Fig. 2D). I chose to run the scaling experiments using a ratio of burrow diameter to earthworm diameter of 0.67, intermediate between the two smallest burrow diameters used in the control experiment. This burrow diameter enabled consistent measurement of near-maximal forces of earthworms that crawled through the burrow several times, enabling not only replicate measurements of force but also measurements of burrowing kinematics.

The two most important properties of a given soil in determining penetration resistance are soil bulk density and water content (Campbell and O’Sullivan, 1991). The objective of the soil control experiments was to choose a soil bulk density and a soil moisture level that would maximize force production by earthworms. In general, animals exert greatest forces when they experience greatest mechanical resistance to force production. Therefore, I hypothesized that earthworms would exert greatest forces when the soil bulk density (mass of solid fraction per total volume of soil) was highest and when the soil moisture was at a level that maximized soil cohesion (which resists deformation). These values depend on the structure and chemistry of the soil used (Campbell and O’Sullivan, 1991) and could not, therefore, be calculated a priori. The soil used was a clay loam (28 % clay, 35 % sand, 37 % silt, ±5 %) in the Tierra soil series (fine, montmorillonitic thermic mollic Palexeralf), collected in Berkeley, CA, USA. I removed the roots and pebbles and dried the soil in a drying oven at 65 ° for 1–3 days. Drying enabled isolation of the solid fraction so that soils of known bulk density and soil moisture could be mixed. The drying and rewetting of soils changes their tensile strength (Whiteley et al., 1981), but new soil was mixed prior to each experiment to maintain consistency in soil properties among experiments.

I measured peak radial force as a 6.1 g earthworm crawled through the open-burrow apparatus configured with three soil bulk densities: 660 kg m⁻³ (as loosely packed as possible while maintaining shape of burrow; large air spaces), 980 kg m⁻³ (moderately packed; small air spaces) and 1300 kg m⁻³ (tightly packed; minute air spaces). The soil bulk densities were achieved by measuring the volume of the space to be occupied by the soil (volume within retaining walls minus volume of burrow; Fig. 1) and calculating the number of grams of water and soil needed to create the desired bulk densities in that volume. All three treatments had the same soil moisture content (0.21 g water g⁻¹ dry soil) and the same ratio of burrow diameter to maximum resting earthworm diameter (0.67). I placed the soil into the apparatus until it was visually homogeneous; earthworm burrowing forces were sufficiently variable that high precision in bulk density homogeneity was not deemed necessary. Five replicate force measurements were recorded for each soil bulk density. Data were collected at 40 scans s⁻¹ at 22 °C in daylight. Peak radial force increased with soil bulk density, but the increase was not statistically significant (Kruskal–Wallis, P>0.10) (Fig. 3A). Nonetheless, the maximum soil bulk density, 1300 kg m⁻³, was chosen for the scaling experiment.

Soil water content was defined on a mass basis (g water g⁻¹ dry soil). I measured peak radial force as a 6.1 g earthworm crawled through the open-burrow apparatus configured with three soil moistures: 0.15 g g⁻¹ (crumbly and incohesive), 0.21 g g⁻¹ (moist and cohesive) and 0.27 g g⁻¹ (muddy). All three treatments had the same soil bulk density (1300 kg m⁻³) and the same ratio of burrow diameter to maximum resting earthworm diameter (0.67). Five replicate force measurements were recorded for each soil moisture. Data were collected at 40 scans s⁻¹ at 22 °C in daylight. Peak burrowing force did not change as a function of soil moisture (Kruskal–Wallis, P>0.10) (Fig. 3B). This result agrees with the observation by Rushton (1986) that burrow lengths of L. terrestris did not significantly correlate with soil moisture content of sample soil blocks. Therefore, the intermediate value 0.21 g g⁻¹ was chosen for the scaling experiments. The soil was exposed to the air intermittently between measurements. The surface of the soil was misted occasionally with water to counteract evaporation, especially in the apparatus assembled for the very small earthworms. These small fluctuations in soil moisture probably had a negligible effect on the force measurements since the soil moisture control experiment demonstrated that similar force measurements were acquired over a great range of soil moisture content.

Earthworms (Lumbricus terrestris L.) ranging in mass from 0.3 to 8.9 g were obtained from Idaho (Loch Lomond Bait, San Rafel, CA, USA) and maintained in the dark at 6 °C in Magic Worm Ranches using Magic Worm Bedding and Magic Worm Food (Magic Products Inc., Amherst Junction, WI, USA). The mature earthworms reproduced in captivity, producing many cocoons. Six of these cocoons hatched, extending the size range down to 0.012 g. Seven size classes of earthworm were chosen for the scaling experiment on the basis of available earthworm sizes and available metal pipe and wire diameters (used to form the burrows), the goal being to maintain a ratio of burrow diameter to maximum resting earthworm diameter of 0.67. The target masses were 0.015, 0.050, 0.20, 0.40, 1.5, 4.5 and 8.0 g. Earthworm mass varied considerably between measurements as a result of the increase and/or decrease in gut contents and body hydration during the day of the experiment. The body mass of each individual was measured to two significant figures on a Mettler balance at least four times: once before measurement of burrowing force, once after measurement of burrowing force and on two other occasions within a few hours before or after the force measurements. The mean mass of each individual was calculated, and an attempt was made to keep this mean within 10 % of the target mass (the mass of an individual earthworm could fluctuate by 20 % during the day).

Muscle area was measured from digital images acquired in a previous study (Quillin, 1998). The digital images show sagittal sections of 20 earthworms (Lumbricus terrestris) ranging in size from 0.16 to 7.5 g. The anesthetized earthworms were frozen in 95 % ethanol chilled to −78 °C using cubes of dry ice. After 2–3 s, the earthworms were removed from the ethanol and cut sagittally with a razor blade to produce two lateral halves (Fig. 4). Images of these halves were captured using a high-resolution color video camera (Sony CCD-Iris SSC-C374) affixed to a Wild Heerbrugg dissecting microscope (model M5A). A RasterOps frame-grabber board was used to select individual video frames. In the present study, I used NIH Image software (Version 1.59) on a Power Macintosh 7100/80 to measure the cross-sectional area of circumferential muscle and the longitudinal area of longitudinal muscle in segments 15 and 50 (Fig. 4). Transverse sections of earthworms were not available for the measurement of the cross-sectional areas of longitudinal muscle. I chose segments 15 and 50 for the measurement of muscle area because these are relatively dissimilar segments. Segment 15 lies in the widest region of the worm near the anterior end, where segments are relatively long and wide and exert large forces on the burrow wall. Segment 50 lies in the relatively homogeneous midregion of the worm, where segments are narrower and shorter and less forceful. Muscle area was measured to the nearest 0.01 mm².

Student’s t distribution was used to test the observed slopes against the hypothesized slopes, and analysis of covariance (ANCOVA) was used to compare observed slopes. All analyses were performed using Systat for Windows (Version 5) on a Power Macintosh 7100/80 computer.

Individual earthworms of all sizes exhibited two distinct force-producing behaviors while burrowing through the open-burrow apparatus: (i) radial anchoring during forward peristaltic locomotion; and (ii) extreme radial expansion to enlarge the burrow, which required a pause in forward locomotion. The radial anchoring forces were represented by small sinusoidal fluctuations in the force record as the earthworm passed through the cylindrical force transducer (Fig. 5A). These small forces were produced as waves of longitudinal muscle contraction (thus radial expansion) moved posteriorly along the length of the earthworm to establish purchase against the burrow wall for locomotion. The magnitude of the radial anchoring forces generally diminished along the length of the body. In contrast, the earthworm typically produced 1–3 spikes of force that were an order of magnitude greater than the anchoring forces. These forces occurred when the earthworms ceased crawling and contracted the longitudinal muscles of several segments simultaneously to enlarge forcefully the diameter of the burrow. These radial forces were exerted mostly by the relatively wide anteriormost 15–20 segments (out of a total of approximately 150 segments), regardless of the size of the earthworm. The maximum radial anchoring force (F_R-anch) and the peak radial burrow-enlarging force (F_R-enlg) were measured for each replicate for each individual.

The earthworms exerted both axial and radial pushes at the end of the dead-end burrow apparatus (Fig. 5B). The pattern of radial and axial force production varied both within and among individuals; sometimes the radial and axial force peaks were in phase, while at other times the radial and axial forces were out of phase. It is not a contradiction to have both forces peak at once; the radial force is produced by the segments within the radial force transducer only, while the axial force can be produced by the segments within or posterior to the radial and axial force transducers. Negative axial forces occurred either when the earthworm everted its pharynx and pulled back (as if to feed) or when suction resisted the removal of the earthworm’s wet, conical ‘nose’ from the conical transducer. The maximum radial anchor force (F_R-enlg) and the peak axial force (F_A) were measured for each replicate for each individual.

All burrowing forces increased as a function of body size, but significantly less than predicted by the null hypothesis. In the open-burrow apparatus, both the peak radial burrow-enlargement force (F_R-enlg) and the radial anchoring force (F_R-anch) scaled as m_b^1/2 rather than the hypothesized m_b^2/3(Fig. 6A). The burrow-enlargement force was an order of magnitude greater than the radial anchoring force for individuals of all sizes. In the dead-end burrow apparatus, both the peak radial burrow-enlargement force (F_R-enlg) and the peak axial force (F_A) also scaled as approximately m_b^1/2 rather than the hypothesized m_b^2/3 (Fig. 6B). On average, the radial and axial forces were comparable in magnitude. The regression statistics for force as a function of body mass are summarized in Table 1.

When the force data were normalized by weight, the results indicated that large earthworms produced less force per newton body weight than small earthworms – even more so than expected by the null hypothesis. All four regressions scaled as approximately m_b^−1/2 rather than the hypothesized m_b^−1/3 (Fig. 7A). Likewise, when the force data were divided by the projected area of application of the force (the plane area of the force transducer perpendicular to the direction of force), the resulting pressures (P) decreased as a function of body mass (P∝m_b^−1/5) (Fig. 7B), whereas the null hypothesis predicted that large and small geometrically similar earthworms would exert the same pressures on their burrows. The regression statistics for force/weight and pressure are summarized in Table 1.

All four measures of muscle area scaled with a mass exponent greater than the 0.67 expected by geometric similarity (Fig. 8).

Earthworms of different sizes were nearly kinematically similar as they crawled through the open-burrow apparatus (Fig. 9). Relative crawling speed (body lengths s⁻¹) varied within and among individuals, but overall did not differ as a function of body mass (t-test, P>0.10). Similarly, relative stride length (stride length/body length) did not differ as a function of body size (P>0.10). However, larger earthworms took fewer strides per second than small earthworms (P=0.007) and used a slightly more elongated body posture (crawling body length/resting body length) than small earthworms (P<0.0001). Burrowing earthworms crawl more slowly, take smaller steps and take fewer steps per time than earthworms crawling on a surface, but the scaling of these variables was similar on the surface and in burrows (Table 2).

The present study tested the null hypothesis that burrowing force is proportional to muscle cross-sectional area and therefore scales as m_b^0.67. This null hypothesis was rejected because pushing force scaled with a mass exponent of 0.45, which was significantly lower than the predicted mass exponent of 0.67. There are several factors that might be responsible for the observation that large worms push with lower force than predicted (or that small worms push with higher force than predicted): (i) muscle area might not increase isometrically with size; (ii) muscle stress might vary with body size; (iii) the mechanical advantage of the earthworm segments might change with body size; (iv) the burrowing kinematics of small and large earthworms might be different; and (v) the resistance of the soil to deformation might depend upon the scale of the deformation.

The null hypothesis assumes that muscle cross-sectional area increases isometrically as a function of body size. Preliminary estimates of the scaling of circumferential and longitudinal muscle area as a function of body size (Fig. 8) indicate that muscle area increases at a greater rate than is expected by isometry. If muscle stress is constant, then a greater relative area of muscle in larger earthworms than smaller earthworms suggests that burrowing force would increase at a greater rate than predicted by isometry, not at a lesser rate as was observed in the present study. Thus, the allometric scaling of muscle area does not appear to account for the observed trend in burrowing force as a function of body size.

The null hypothesis assumes that muscle stress (the contractile force exerted per cross-sectional area of muscle) is independent of body size. The muscle stress of vertebrate striated muscle appears to change little as a function of body size when different species are compared and also when organisms of different ages within a species are compared (Close, 1972). The scaling of muscle stress in earthworm obliquely striated muscle is not known. However, since force decreases and relative muscle area increases as a function of body size, one testable hypothesis is that muscle stress decreases as a function of body size during earthworm ontogeny.

The muscle force transmitted to the environment will be proportional to the muscle stress only if the muscle cross-sectional area scales isometrically and the mechanical advantage of the system does not change as a function of body mass (because mechanical advantage = F_out/F_in = L_in/L_out, where F is the force and L is the lever arm; see Alexander et al., 1981; Biewener, 1989). Since mechanical advantage is determined by the geometry of the system, mechanical advantage is either implicitly or explicitly assumed to be constant as a function of body mass in geometrically similar organisms. Mechanical advantage has yet to measured in hydrostatic skeletons that lack lever arms, but it is hypothesized that mechanical advantage is determined by the length-to-diameter ratio of constant-volume segments (see Chapman, 1950; Arthur, 1965; Kier and Smith, 1985). Earthworm segments grow with a constant length-to-diameter ratio (Quillin, 1998), suggesting that a change in mechanical advantage as a function of body mass does not account for the observed relationship between force and body mass. Although relative muscle area is not constant as a function of body mass, muscle area is a determinant of F_in rather than mechanical advantage per se (e.g. when the biceps of a bodybuilder atrophy, mechanical advantage stays constant, even though force production decreases).

Quillin (1999) measured the scaling of kinematics of earthworms crawling on a surface and concluded that earthworms (0.012–8.5 g) were essentially kinematically similar. The kinematics of burrowing earthworms appear to scale similarly (Table 2), although the absolute magnitudes of crawling speed, stride length and stride frequency are higher in surface crawling than in burrowing. One statistically significant result was that larger earthworms tend to burrow with a more elongated body than smaller earthworms. For example, a 8.9 g worm burrows with a ratio of burrowing body length to resting body length of 1.13, whereas a 0.15 g worm burrows with a ratio of burrowing body length to resting body length of 0.83, a difference of 35 %. Obliquely striated muscle tends to have a flatter force/extension curve than cross-striated muscle (Full, 1997) such that maximum force is not as highly dependent upon muscle strain. The length/tension curve for earthworm muscle generated by Hidaka et al. (1969) supports this generalization. However, their data suggest that a 35 % change in muscle length may result in a comparable change in muscle tension. Thus, one testable hypothesis is that larger earthworms exert lower forces than expected because their muscles are working at a higher strain, where the ability to exert force is diminished.

Pores in soils can range in size over several orders of magnitude and generally constitute 30–60 % of the volume of soil (Marshall and Holmes, 1988). Since pores are one important determinant of the mechanical properties of soil, the properties of soil are different at different scales (soil scientists use penetrometers of different sizes to measure the strength of soil at different scales; Campbell and O’Sullivan, 1991). Although either metal or Plexiglas served as the interface between the earthworms and the force transducers in the present experiment, the scaling of soil properties may have had an indirect effect on the forces that the earthworms could exert (e.g. by affecting how well worms of different sizes could anchor in the soil burrow). This hypothesis remains to be tested and has important ecological implications.

A recent study reports maximum axial and radial pressures exerted by L. terrestris in Plexiglas tubes (Keudel and Schrader, 1999). The mean axial pressure of 51 individuals (unknown size) was 46.26±2.95 kPa (mean ± S.E.M.) while the mean radial pressure of nine individuals (unknown size) was 72.62±12.46 kPa (mean ± S.E.M.). The values of Keudel and Schrader (1999) are higher than those reported in the present study (Fig. 7; assuming that their individuals were large juveniles and/or adults). This difference may be due to the condition of the study animals (e.g. level of hydration) or to the method of force measurement (e.g. ratio of worm diameter to diameter of force transducer). Keudel and Schrader (1999) also observed that radial pressures were greater in magnitude than axial pressures, whereas the present data reported no difference. It is not clear whether the present study underestimated radial pressure, or whether Keudel and Schrader (1999) underestimated axial pressure. One methodological difference between the two studies concerns the shape of the depression against which the earthworms pushed axially. Keudel and Schrader (1999) used a rectilinear tip, whereas I used a conical tip. Since the prostomium of earthworms is very sensitive, this methodological difference may have had an effect on pushing pressure.

Soft-bodied organisms with hydrostatic skeletons generally appear to move more slowly than their counterparts with rigid skeletons, with some notable exceptions (Maitland, 1992; Kier and Van Leeuwen, 1997; Brackenbury, 1997). Also, the cost of transport of organisms with hydrostatic skeletons appears to be higher than for arthropods of the same body mass (Casey, 1991; Berrigan and Lighton, 1993). How do forces exerted on the environment compare? The force/weight exerted by adult earthworms is intermediate compared with the force/weight exerted by wedging ground beetles (Evans 1977; Evans and Forsythe, 1984), righting cockroaches (Full et al., 1995) and load-carrying rhinoceros beetles (Kram, 1996) of similar size (Fig. 10). Compared with other hydrostatic skeletons, the burrowing forces of adult Lumbricus terrestris were smaller than those exerted by burrowing caecilians (legless amphibians) of twice the length (mass not provided, O’Reilly et al., 1997) and comparable in magnitude with the forces measured in a smaller earthworm, Aporrectodea rosea (McKenzie and Dexter, 1988a,b). Maximum forces vary widely, even among closely related species (Evans and Forsythe, 1984; Keudel and Schrader, 1999). Overall, it appears that force-generating ability depends upon the subtle mechanical design and ecological strategy of the organism (e.g. fast crawler versus forceful burrower), not whether the skeleton is hydrostatic or rigid.

Most scaling studies have focused on comparisons of adults of different species (e.g. Evans, 1977; Alexander, 1985; Biewener, 1989) rather than an ontogenetic size range. Of the ontogenetic studies, the measurements tend to be of performance variables such as sprint speed (Garland, 1985; Marsh, 1988), maximum jump distance (Queathem, 1991) and jump acceleration (Carrier, 1995). Inferences about forces can be made from these studies but, overall, there are few data for the comparison of ontogenetic force production in hydrostatic skeletons versus rigid skeletons.

For two organisms of different sizes to be dynamically similar, the ratios of linear dimensions, velocities and forces must be the same (Daugherty and Franzini, 1997). Quillin (1998) determined that Lumbricus terrestris grows with geometric similarity, satisfying the first requirement for dynamic similarity. Quillin (1999) observed that L. terrestris maintains kinematic similarity while crawling on a surface, and the present study indicates kinematic similarity in burrowing as well, satisfying the second requirement of dynamic similarity. The present study has demonstrated that the ratios of one burrowing force (e.g. radial) to another burrowing force (e.g. axial) remain constant as earthworms grow. However, the ratio of burrowing force to gravitational force (weight) decreases as a function of body size. Burrowing force and weight are only two components of the total balance of forces on the burrowing earthworm (for the equation of motion of crawling earthworms, see Keller and Falkovitz, 1983), but the data on these two forces seem to suggest that earthworms do not possess dynamic similarity in burrowing.

In summary, larger earthworms exert greater absolute burrowing forces than smaller earthworms, but the difference is less than expected on the basis of traditional scaling theory derived for jointed skeletons. Testable hypotheses explaining this discrepancy include: (i) that muscle stress decreases during ontogeny, (ii) that larger earthworms exert smaller forces than expected because their muscles work at larger strains, and (iii) that the scaling of soil properties affects the scaling of burrowing forces in earthworms. Earthworm forces fall within the range of forces observed for other animals of comparable mass, but there are few ontogenetic force regressions in other studies to provide comparison and contrast to the present study.

This research was supported by a NSF Graduate Research Fellowship, a University of California at Berkeley Frankhauser Fellowship and a Sigma Xi Grants-in-Aid of Research to K.Q, by NSF Grant 92-20525 to M. Koehl and R. Keller and by ONR Grant 444095-23068 to M. Koehl. I am grateful to M. Koehl for helpful discussion and advice, to G. Wang for assistance with the force experiments and to R. Amundsen for his help in classifying the soil. I am also grateful to M. Koehl and R. Kram for critical reading of this manuscript and to the members of the biomechanics group at the University of California at Berkeley for their insights and suggestions.