Ontogenetic Scaling Of Hydrostatic Skeletons: Geometric, Static Stress And Dynamic Stress Scaling Of The Earthworm Lumbricus Terrestris

Body size influences almost every aspect of the biology of an organism, from its physiology and ecology to the mechanical functioning of its skeleton (reviewed in Gould, 1966; Currey, 1970; Alexander, 1971; Pedley, 1977; McMahon, 1973, 1975; McMahon and Bonner, 1983; Peters, 1983; Calder, 1984; Schmidt-Nielsen, 1984). Studies on the scaling of skeletons have concentrated on vertebrates, arthropods and trees whose rigid bones, exoskeletons and trunks, respectively, are loaded as beams and columns by their own body weight. Other studies have examined the scaling of structures loaded in pure tension (e.g. fruit stems and kelp stipes; Peterson et al. 1982; Johnson and Koehl, 1994). While these studies have established a foundation of useful scaling principles, they are not directly applicable to the problem of how hydrostatic skeletons scale.

Soft-bodied organisms with hydrostatic skeletons are abundant and diverse. Body mass differs by at least 13 orders of magnitude from giant squid (Architeuthis sp., 20 m long including tentacles) to minute nematode worms (less than 1 mm length) (Ruppert and Barnes, 1995). Body size also increases by orders of magnitude during the growth of many soft-bodied organisms, for example by four orders of magnitude in body mass in the earthworm Lumbricus terrestris. Clearly, size is an important variable for soft-bodied organisms, but what effect does body size have on the biomechanical functions of hydrostatic skeletons?

The hydrostatic skeletons of most soft-bodied organisms are constructed from an extensible body wall in tension surrounding a fluid or deformable tissue under compression (Chapman, 1958; Wainwright, 1988). The fluid under compression becomes pressurized, and it is this pressure (rather than the rigid levers of vertebrates and arthropods) that enables stiffening of the organism, antagonism of muscles and transfer of muscle forces to the environment (e.g. Chapman, 1958; Currey, 1970; Trueman, 1975). Hydrostatic skeletons may encompass whole organisms, either for an entire lifetime (e.g. earthworms) or for one life-history stage (e.g. caterpillars), or alternatively may occur in just parts or appendages of an organism (e.g. tongues in vertebrates and tube feet in echinoderms). Hydrostatic appendages whose requisite incompressible fluid is contained within deformable muscle cells are called muscular hydrostats (e.g. elephant trunks and squid tentacles; Kier and Smith, 1985). In contrast, hydrostatic skeletons describe a body cavity filled with sea water, blood, coelomic fluid and/or deformable organ tissues (e.g. sea anemones, roundworms and earthworms; see Kier, 1992).

How does a soft-bodied organism grow by orders of magnitude in body size yet maintain the biomechanical functions of its hydrostatic skeleton? The objectives of the present study were (1) to generate the geometric, static stress and dynamic stress similarity scaling hypotheses for hydrostatic skeletons on the basis of their form and function, and (2) to apply these similarity hypotheses to the ontogeny of the earthworm Lumbricus terrestris to determine which aspects of skeletal function are conserved or altered as a function of size during growth.

Whether a particular functional aspect of a skeleton is conserved or altered in relation to body mass is typically expressed in terms of a scaling hypothesis. Some hypotheses are based on form (‘geometric similarity’) whereas others are based on function (e.g. ‘stress similarity’). The functional similarity hypotheses have been defined previously in terms of the mechanism of mechanical function of lever-like skeletons. Since hydrostatic skeletons function by a qualitatively different mechanism from lever-like skeletons, both the predictions of the similarity hypotheses and their relationships to one another must be derived anew.

If two organisms are geometrically similar, or isometric, then linear dimensions are proportional to volume raised to the one-third power. Linear dimensions scale as (body mass)^1/3 because mass is proportional to volume which is proportional to the cube of length, when body density is constant (e.g. Thompson, 1917; Alexander, 1971; Schmidt-Nielsen, 1984; Fig. 1). The exponential relationship between linear dimensions and body mass can be expressed by the function y=am_b^b (Huxley, 1932), where m_b is body mass, y is a linear dimension, a is a constant and b is the exponent (1/3 in the case of geometric similarity). If the exponent is significantly different from 1/3, scaling is said to be allometric.

Most hydrostatic skeletons are cylindrical in shape, having an approximately round or elliptical cross section and an easily identifiable longitudinal axis (Wainwright, 1988). Therefore, the most important linear dimensions are length (L), diameter (d) and body wall thickness (t), all of which scale as m_b^1/3 in geometrically similar organisms in comparable postures, such as resting posture. If the cylindrical body of a hydrostat is divided into segments (as in annelids), then the number and linear dimensions of segments scale as m_b⁰ and m_b^1/3, respectively, in geometrically similar organisms.

Static stress is the force (F) per cross-sectional area (A) of a skeletal element bearing the force when an organism is standing still (Thompson, 1917; Hill, 1950; McMahon, 1975). Static stress similarity occurs when static stress is constant as a function of body mass (i.e. stress ∝ m_b⁰). The loading of beam-like skeletons by body weight precludes the possibility that they are both geometrically and statically stress similar. Weight is proportional to the cube of the linear dimension, while the cross-sectional area of a skeletal element is proportional only to the square of the linear dimension. Therefore, static stress in a self-loaded beam can only remain constant as a function of body size if diameter increases at a greater rate than length (e.g. McMahon, 1975; Fig. 1). Bending beams must scale allometrically, not geometrically, to maintain static stress similarity.

Unlike the case for rigid skeletons, static stress similarity is not excluded by geometric similarity in hydrostatic skeletons, and vice versa; hydrostats that are geometrically similar (and therefore possess the same r/t ratio) may or may not show static stress similarity depending on the magnitude of internal pressure. Likewise, hydrostats that are not geometrically similar may or may not show static stress similarity depending on the magnitude of internal pressure.

Animals experience dynamic stresses in their skeletons when they move. The dynamic strain similarity hypothesis of Rubin and Lanyon (1984; so-named because they calculated stresses from their measurements of bone strains) states that the maximum stress experienced by a skeletal element during vigorous locomotion is independent of body size and thus scales as m_b⁰ (Fig. 1). Maximum stress during locomotion in legged animals is greater than resting stress owing to the addition of the ground reaction forces (F_acc) required to accelerate and decelerate the animal. The magnitude of ground reaction forces is dependent not only on the weight of the organism but also on the gait it uses. Therefore, the dynamic strain similarity hypothesis as applied to rigid skeletons is neither excluded nor predicted by geometric similarity alone; it must be measured while the organisms are in motion.

The maximum stress experienced by the body wall of a hydrostatic skeleton likewise depends on the organism’s behavior. The internal pressures in most hydrostats fluctuate (Zuckerkandl, 1950), but tend to peak when muscle contractions peak (e.g. Chapman and Newell, 1947; Newell, 1950). However, hydrostats tend to be highly deformable, and the maximum r/t may not coincide with the pressure maximum. The maximum tensile stress occurs when and where the product of P and r/t is greatest (see equation 1). If the main source of internal pressure is muscle tension (P_m, which is likely to be the case) and if muscle properties do not vary with body size, then geometrically similar hydrostats should show dynamic stress similarity as long as behavior does not change during growth. Thus, internal pressure and body wall geometry must be measured in living hydrostats in order to test hypotheses of dynamic stress similarity.

The first objective of the present study was to generate predictions for the geometric, static stress and dynamic stress scaling hypotheses for hydrostatic skeletons on the basis of their form and function. Given the formulated predictions, the next objective was to apply the similarity hypotheses to the ontogeny of the earthworm Lumbricus terrestris to determine which aspects of skeletal function are conserved or altered as a function of size during growth. L. terrestris was chosen as the experimental system because of its abundance, ease of study in the laboratory, and the availability of a large ontogenetic size range. Its segmented skeleton is composed of a body wall (cuticle, epithelium, two layers of muscle and connective tissue) surrounding coelomic fluid and deformable organ tissues. The volume of coelomic fluid within each segment is constant (Newell, 1950). Thus, when the circumferential muscles of a segment contract, the segment becomes long and slim as the passive longitudinal muscles are stretched. When the longitudinal muscles contract, the segment becomes short and wide and the circumferential muscles are stretched (Seymour, 1969). Alternating waves of circumferential and longitudinal muscle contractions travel posteriorly along the body, enabling forward progression by peristalsis (e.g. Gray and Lissmann, 1938).

The experimental objectives of the present study were (1) to measure the resting geometry of the skeletons of earthworms as a function of size, (2) to measure internal pressures in resting earthworms in order to calculate the scaling of static tensile stresses in the body wall, and (3) to measure dynamic internal pressures in locomoting earthworms in order to calculate the scaling of dynamic tensile stresses in the body wall. The present study focuses on the earthworms during their most observed and understood activity – surface crawling – although peristalsis is also used by this species to burrow. Whereas most earthworm species are either geophagous (earth-eaters) or surface feeders, Lumbricus terrestris is both; as its common name ‘night crawler’ suggests, it feeds on the surface at night, drawing leaves and other organic materials into its semi-permanent burrows and digesting them within (Darwin, 1881; Arthur, 1965; Satchell, 1967). Thus, surface crawling is used as a pragmatic first assessment of dynamic stresses in the body wall.

Earthworms (Lumbricus terrestris L.) ranging in size from 0.3 to 8.0 g were obtained from Idaho (Loch Lomond Bait, San Rafel, CA, USA) and maintained at 6 °C in Magic Worm Ranches using Magic Worm Bedding and Magic Worm Food (Magic Products Inc., Amherst Junction, WI, USA). I supplemented this size range with juveniles (0.01–0.3 g) collected from an outdoor enclosure in Berkeley, CA, USA, in which I maintained L. terrestris from Canada (Berkeley Bait, Berkeley, CA, USA). Fig. 3 summarizes the size ranges used for each of the scaling variables examined.

Although earthworm segments are similar along the length of the body, some regionalization does occur. I chose two relatively dissimilar segments for comparison of diameter, body wall thickness, internal pressure and stress to assess the effects, if any, of this regionalization on the function and scaling of the hydrostatic skeleton. Segment 15 lies in the widest region of the worm near the anterior end, where segments are relatively long and wide and contain specialized digestive and reproductive organs. Segment 50, by contrast, lies in the relatively homogeneous midregion of the worm, where segments are narrower and shorter and are occupied primarily by the intestine (Fig. 4).

I made continuous measurements of internal pressure using a low-volume gauge pressure transducer (PX170, Omega Engineering, Stamford, CT, USA) designed to read pressures up to 7 kPa. I connected the transducer to a piece of water-filled flexible polyethylene tubing attached to a terminal cannula constructed from a 27 gauge hypodermic needle. The transducer was calibrated before each series of measurements by inserting the needle through the wall of a 2 cm vertical Nalgene tube that was open at the top. The tube height was raised so that the meniscus of the water level increased in height at 1 cm intervals with respect to the center of the transducer. The data were collected at 50 scans s⁻¹ by LabVIEW software (version 3.0.1; National Instruments, Austin, TX, USA) on a Gateway 2000 computer.

Prior to insertion of the cannula, the earthworms were calmed by placement in dilute ethanol (1–5 % in spring water) for 20–40 min (response time to the anesthetic varied). After this procedure, worms did not thrash when handled but did exhibit typical locomotory peristalsis as observed in earthworms without a cannula inserted. I then placed the earthworms into a tray of fresh water 1 cm deep (meniscus level adjusted to the height of the center of the pressure transducer) rather than in air to keep the needle-tip submerged between measurements and to keep the tubing still while the worm crawled ‘in place’ in the water, avoiding pressure artifacts due to movement of the apparatus. I inserted the needle laterally into the coelom of each worm, just under the body wall in segments 15 and 50 or the adjacent segments (pressure among adjacent segments did not differ measurably). The following variables were measured from the pressure records (to the nearest 10 Pa; Fig. 5): (1) maximum pressures during circumferential muscle contractions (P_C.M.), (2) maximum pressures during longitudinal muscle contractions (P_L.M.) and (3) resting pressures (P_R). The dynamic pressure measurements were averaged over 10 cycles of peristalsis. Resting pressures were recorded as often as possible but only as they occurred between bouts of peristalsis (rather than in anesthetized or otherwise quiescent earthworms) to ensure that the cannula was not clogged. The number of pressure records available for each individual varied from zero to five; the mean resting pressure was calculated when more than one record was available.

I anesthetized the earthworms to compare the dimensions of their highly extensible skeletons in comparable postures. First, I submerged the earthworms in spring water and slowly added drops of 50 % ethanol for 30–60 min. When the earthworms no longer responded to stimulation, I blotted them dry and measured the mass of the large and medium worms on a three-beam balance and the smallest worms on a Mettler balance (model AE 163) to two significant figures (i.e. to the nearest 0.1 g for the largest worms and to the nearest 0.0001 g for the smallest worms). I then counted the number of segments and measured body length with a ruler or digital calipers to the nearest 1 mm (or 0.1 mm in the case of very small worms) and the widths of segments 15 and 50 with calipers to the nearest 0.1 mm.

Body wall thickness was measured using frozen sections rather than standard histological sections to minimize shrinkage due to fixation. I rapidly froze anesthetized earthworms by sliding them off a straight edge into 95 % ethanol chilled to −78 °C using cubes of dry ice. Small worms froze immediately while large worms required 2–3 s; body dimensions did not change measurably during this period. I removed each frozen worm and cut it first transversely with a razor blade to remove two sections composed of segments 13–17 and 48–52, and then sagittally to produce two lateral halves of each section (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). I used a RasterOps frame-grabber board to select individual video frames, and NIH Image software (Version 1.59) on a Power Macintosh 7100/80 to measure body wall thickness in the middle of segments 15 and 50 and at the annuli adjacent to these segments (Fig. 4).

Thickness was measured to the nearest 0.01 mm, and dorsal and ventral thicknesses were averaged. Segment length was measured to the nearest 0.01 mm. Circumferential muscle cross-sectional area was measured to the nearest 0.01 mm². The lower size limit of worms sectioned (Fig. 3) was determined by the lack of pigmentation in small worms, which rendered tissue layers difficult to distinguish.

Static body wall tensile stresses were calculated from equation 1 using resting pressure and the resting radius and thickness values obtained from morphometric measurements of the same individuals. Since the body walls of earthworms do not have uniform thickness (Fig. 4), the annulus body wall thickness was used to calculate maximum resting longitudinal stress (because only the longitudinal muscles bear longitudinal loads). The mid-segment thickness was used to calculate resting circumferential stress (because both longitudinal and circumferential muscles bear circumferential loads and because circumferential stress is resisted by the muscular septae at each annulus).

Dynamic body wall tensile stresses were calculated using pressures obtained during maximum contractions of circumferential and longitudinal muscles and using radius and thickness values calculated from the resting dimensions of the segments and the maximum shape changes observed in the segments during peristalsis. I video-taped the shape changes of segments 15 and 50 of crawling earthworms through a Wild Heerbrugg dissecting microscope using a SONY CCD-Iris (SSC-C374) high-resolution color video camera and a time/date generator (Panasonic WJ-810). I used RasterOps Video Capture software on a Power Macintosh 7100/80 to grab video frames, and NIH Image software (Version 1.59) to measure the minimum and maximum length and lateral diameter of the segments to the nearest 0.1 mm. During longitudinal muscle contraction, the shapes of segments 15 and 50 did not differ from those of resting earthworms; therefore, the resting radius and thickness values were used for calculation of circumferential stress during longitudinal muscle contraction. During circumferential muscle contraction, the length of segments 15 and 50 of earthworms of all sizes increased by a factor of 1.6±0.2 (N=20), while the diameter decreased by a factor of 0.75±0.06 (means ± S.D., N=15) (K. J. Quillin, unpublished data). Using these values, I solved for the annulus thickness of the extended segments assuming that the volume of the body wall is constant.

Student’s t distribution was used to test slopes where the null hypothesis was β=0 or β=1/3. Analysis of covariance (ANCOVA) tests of homogeneity of slopes were performed to test whether observed slopes were significantly different from each other (Sokal and Rohlf, 1969). Independent-sample and paired-sample t-tests were performed to compare internal pressures in segment 15 versus segment 50. All statistical analyses were performed using Systat for Windows, version 5. Values are given as means ± standard deviations (S.D.).

The external dimensions of the earthworms increased isometrically. Segment number (n) varied among individuals (147±13, N=111), but did not increase with body mass m_b over the two orders of magnitude size range investigated (n∝m_b^0.01; the slope was not significantly different from zero, P>0.10; Fig. 6), indicating that the earthworms grew by enlarging each segment rather than by adding segments. Furthermore, the resting lengths L_s of segment 15 (L_s=0.011m_b^0.32; r²=0.82, N=19) and segment 50 (L_s=0.007m_b^0.34; r²=0.79, N=19) increased at the rate predicted by geometric similarity (the slopes were significantly different from zero, P<0.001, but not significantly different from 0.33, P>0.10). The resting diameters (d₁₅, d₅₀) of both segments also increased as predicted by geometric similarity (d₁₅∝m_b^0.34, d₅₀∝m_b^0.32; P=0.061 for both compared with a slope of 0.33, P<0.001 for both compared with a slope of zero; Fig. 7), indicating that earthworms increased their segment dimensions isometrically. The resting length of the entire body increased with body mass at the same rate as predicted by geometric similarity (L∝m_b^0.34; P>0.10 compared with a slope of 0.33, P<0.001 compared with a slope of zero, N=164; Fig. 7). The mean length/diameter ratio over the entire size range was 21.8±4.0 (N=135).

Body wall thickness increased isometrically in segment 15, but allometrically in segment 50 (Fig. 8). The mid-segment and annulus thicknesses of segment 15 increased isometrically as m_b^0.37 and m_b^0.38 respectively (P>0.10 compared with a slope of 0.33, P<0.001 compared with a slope of zero). However, the mid-segment and annulus thicknesses of segment 50 grew allometrically as m_b^0.42 (P=0.004 compared with a slope of 0.33) and m_b^0.45 (P<0.001), respectively. The allometric increase in body wall thickness reflects an allometric increase in the cross-sectional area of the muscles; the mid-sagittal cross-sectional areas of circumferential muscle in the body wall of segments 15 and 50 increased as m_b^0.90 (r²=0.90, N=20) and m_b^0.91 (r²=0.81, N=20), respectively, which were both greater than the slope 0.67 predicted by isometry (P=0.006 for segment 15, P=0.037 for segment 50).

Despite allometry in body wall thickness, the longitudinal and circumferential muscles maintained their relative proportions during growth (Fig. 8). The slopes of the mid-segment body wall thicknesses (a measure of circumferential muscle plus longitudinal muscle plus epidermis and cuticle thickness) were not significanly different (P>0.10) from the slopes of the annulus body wall thicknesses (a measure of longitudinal muscle plus epidermis and cuticle thickness; see Fig. 4).

Resting pressures P_R differed greatly from individual to individual (115±45 Pa; N=19), but scaled independently of body mass over the one-order-of-magnitude range in body mass observed (P_R∝m_b^−0.003; P>0.10 compared with a slope of zero; Fig. 9). Resting pressures in segments 15 and 50 were not significantly different (independent t-test, P>0.10), so these data were pooled for the calculation of stress.

Static tensile stress in the body wall of the earthworms was constant as a function of body mass. The static circumferential stress σ_c in the mid-segment body wall scaled as m_b^−0.07 and m_b^−0.09 in segments 15 and 50, respectively, neither of which was different from a slope of zero (P>0.10; Fig. 10). The static longitudinal stress σ_l in the longitudinal muscle layer scaled as m_b^0.10 and m_b^0.07 in segments 15 and 50, respectively, neither of which was significantly different from a slope of zero (P>0.10; Fig. 9). The mean static circumferential stresses were 830±330 Pa and 710±300 Pa (N=19) for segments 15 and 50, respectively. The mean static longitudinal stresses were 64±15 Pa and 55±13 Pa (N=19) for segments 15 and 50, respectively.

Since the mean internal pressures during maximum circumferential muscle contraction P_C.M. in segments 15 (580±220 Pa) and 50 (670±300 Pa) were not significantly different (P>0.10), these data were pooled for the calculation of stress. However, the internal pressures during maximum longitudinal muscle contraction P_L.M. in segments 15 (300±220 Pa) and 50 (550±260 Pa) were significantly different (P=0.018), and so were not pooled for the calculation of stress. Maximum dynamic stresses in the body wall during peristalsis were independent of body mass (Fig. 10). The mean maximum pressure during circumferential muscle contraction P_C.M. scaled as m_b^0.04 (Fig. 9), and the resulting maximum longitudinal stress σ_l in the body wall scaled as m_b^0.01 and m_b^−0.19 in segments 15 and 50, respectively (Fig. 10). The mean maximum pressure during longitudinal muscle contraction P_L.M. scaled as m_b^−0.05 (Fig. 9), and the resulting maximum circumferential stress σ_c in the body wall scaled as m_b^−0.07 and m_b^−0.09 in segments 15 and 50, respectively. None of the regression slopes was significantly different from zero (P>0.10), indicating that earthworms show dynamic stress similarity in peristaltic crawling as they grow. The amplitude of stress fluctuation during peristalsis was higher in segment 15 than in segment 50 (P<0.001; Fig. 10), but the amplitude did not change significantly as a function of body mass (P>0.10).

In the resting earthworm, circumferential tensile stress was greater than longitudinal tensile stress (Fig. 10). In the crawling earthworm, however, longitudinal tensile stress during circumferential muscle contraction was greater than circumferential tensile stress during longitudinal muscle contraction as a result of the higher pressures generated during circumferential muscle contraction (Fig. 9) and smaller cross-sectional area of the body wall experiencing longitudinal forces (Fig. 4).

The highest pressures recorded in the present study occurred during violent, whole-body contractions rather than during normal peristaltic crawling. Pressures reached 6.2 kPa in a 5.5 g earthworm and 4.3 kPa in a 1.8 g earthworm. The resulting circumferential stresses were calculated as 44 and 33 kPa, respectively.

The earthworm Lumbricus terrestris grows isometrically while maintaining static and dynamic stress similarity. Unlike rigid skeletons, for which geometric similarity and static stress similarity are mutually exclusive hypotheses, hydrostatic skeletons can simultaneously maintain similarity in both form and this aspect of mechanical function as they grow.

The main source of loading on the skeleton of most terrestrial organisms with rigid skeletons is body weight. In earthworms, the main source of loading on the skeleton is internal pressure (generated by body wall muscles contracting against a constant volume of internal fluid). The pressure measurement apparatus used in the present study necessitated submerging the earthworms in a shallow bath of water to prevent air from entering the system and to prevent pressure artifacts due to the movement of the apparatus. Thus, gravitational pressure (P_g) was essentially eliminated as a source of total internal pressure. A horizontally oriented earthworm that is 1 mm in dorsoventral diameter should have a dorsoventral gradient in pressure that ranges from zero at the dorsal perimeter to a maximum P_g of 10 Pa at the ventral perimeter. An earthworm that is 10 mm in dorsoventral diameter should have a pressure gradient that ranges from zero to a maximum P_g of 100 Pa. These pressures lie within the variation in resting pressure measured in the present study (Fig. 9), and thus would not affect the resting pressure regression appreciably. Furthermore, the estimated gravitational pressures are considerably smaller than the pressures that occurred during peristalsis and therefore appear to be relatively unimportant. Vertically oriented earthworms (e.g. in vertical burrows) would not necessarily have higher gravitational pressures than horizontally oriented earthworms since muscular septae divide the coelom into constant-volume segments which prevent the transfer of pressure along the length of the body. Overall, the decoupling of body weight and skeletal function probably accounts to a great extent for the ability of a terrestrial hydrostatic skeleton to grow isometrically while maintaining stress similarity.

Lumbricus terrestris is generally considered to be a large earthworm compared with other temperate species (e.g. Arthur, 1965; Piearce, 1983), but several ‘giant’ species, measuring meters in length, can be found in the tropics (Stephenson, 1930). The upper limit to the size of hydrostatic skeletons is unclear, but some of the possible limitations to giant earthworms include (1) a decreased respiratory surface area due to the low surface-to-volume ratio compared with that of smaller earthworms, (2) an increased importance of gravitational pressure as a source of load on the body wall, (3) an increased frictional resistance to burrowing, and (4) the exponential increase in the cost of tunnel construction with increasing body diameter (Gans, 1973). Present evidence suggests that larger earthworm species generally possess a higher length/diameter ratio and more segments than smaller earthworm species (Arthur, 1965; Piearce, 1983); this allometry may enable earthworms to reach the upper limit of body mass for terrestrial hydrostatic skeletons.

Segments are the constant-volume functional units of the skeletons of earthworms. Some species add segments during post-emergent growth (Evans, 1946; Piearce, 1983), but Lumbricus terrestris possesses the same number of segments throughout its lifetime (unless segments are lost as a result of predation; Evans, 1946). The number of segments and their dimensions are important for several reasons. In general, the greater the degree of segmentation of the skeleton, the greater the potential for localization of forces, pressures and shape changes in the skeleton and, hence, the greater the potential for complex motions (Clark, 1964). Furthermore, since the muscular septae between segments radially reinforce the cylindrical structure, the number of segments for a given body length may affect the resistance of the body to circumferential bulging and sagging (e.g. Seymour, 1970). A large number of septae per body length may also diminish the pressure gradient established along the length of an earthworm oriented vertically in its burrow.

The number and dimensions of the segments are also important in determining the velocity advantage of a hydrostatic skeleton. The velocity advantage is expressed by the ratio U₂/U₁, where U₁ is the rate of contraction of a muscle and U₂ is the maximum resulting velocity of an attached skeletal element (e.g. Alexander, 1983). In vertebrates and arthropods, the amplification of velocities is accomplished by rigid levers rotating about pivots. The velocity advantage of hydrostatic skeletons is determined by the simple geometric relationship between length and diameter in a constant-volume deformable cylinder (Fig. 11; Chapman, 1950; Kier and Smith, 1985). Both segments 15 and 50 have resting length/diameter ratios less than 1. This means, for example, that only a slight shortening of the longitudinal muscles is necessary for a large and rapid increase in diameter, enabling the earthworm to anchor quickly. Since the length and diameter of segments 15 and 50 did not differ significantly as a function of body mass, the segments of large and small worms appear to possess the same velocity advantage. Velocity advantage and mechanical advantage tend to be inversely related, but mechanical advantage has yet to be measured in hydrostatic skeletons.

The length/diameter ratio of an earthworm is not only important to the mechanical function of the skeleton, but also to the physiology of the organism. Gaseous exchange in earthworms occurs by diffusion across the skin (for a review, see Edwards and Bohlen, 1996), and the rate of water loss is also likely to be proportional to body surface area (Piearce, 1983). Since the surface area of L. terrestris decreases relative to its body mass during ontogeny (growth is isometric; thus, surface area ∝ m_b^2/3), either larger earthworms have a lower respiratory rate and activity level than smaller earthworms or the decrease in relative surface area is compensated by other physiological variables such as the degree of vascularization of the body wall or the oxygen-carrying capacity of hemoglobin.

Whereas I observed isometric growth in L. terrestris (L∝d^1.05), Piearce (1983) observed an increase in the length/diameter ratio as a function of body mass (L∝d^1.28) in preserved specimens of the same species, such that larger worms were skinnier relative to their length than were smaller worms. All of the individuals observed by Piearce (1983) possessed a lower length/diameter ratio (L/d≈18) than I observed in the present study (L/d≈22). Preservation artifacts or interpopulational differences in diet and habitat are possible sources of this discrepancy.

Neither resting pressure nor peak pressures during peristalsis changed as a function of body mass, and values were comparable to, but on average lower than, those reported for adult worms of the same species by Seymour (1969; mass range 4–6 g) and Newell (1950; mass range not provided). Since both body geometry and internal pressures were essentially independent of body mass in L. terrestris, I predict that the maximum force exerted on the environment by an earthworm per area of force application will be constant as a function of body mass (F/A_f∝m_b⁰), providing that muscle stress is constant. However, given the allometric increase in body wall thickness of segment 50, maximum internal pressures and thus the maximum force exerted by the mid-region of large earthworms may be greater than that in small earthworms during burrowing. I am currently testing these alternative hypotheses.

One might expect internal pressure to peak during both circumferential and longitudinal muscle contractions, since these muscles are said to antagonize one another via internal pressure (e.g. Chapman, 1950). In the present study, internal pressure peaked during circumferential muscle contraction but decreased (sometimes nearly to resting pressure) during longitudinal muscle contraction (Fig. 5). Seymour (1969) hypothesized that the body wall, which is stiffened by the collagen-fiber-reinforced cuticle and subepidermal connective tissue, may act as a spring, storing potential energy upon circular muscle contraction which can be released to aid the longitudinal muscles in shortening upon the relaxation of the circumferential muscles. This spring-like behavior of segments may also be a simple byproduct of the dynamic shape changes of the constant-volume segments. Clark and Cowey (1958) established that a unit length of an open-ended cylinder bounded by crossed-helical collagen fibers contains the greatest volume when the angle between the fibers and the longitudinal axis of the cylinder is 54°44′ (see Fig. 1 in Clark and Cowey, 1958). The fiber angle of an adult earthworm cuticle is approximately 45 ° when the animals are at rest (Richards, 1974; Lepescheux, 1988). Since the volume of a segment is constant, the volume cannot decrease when the circumferential muscles contract (causing the segment to become longer and slimmer and the fiber angle to decrease), resulting in an increase in internal pressure. Likewise, the volume of a segment cannot increase when the longitudinal muscles contract (causing the segment to become short and wide and the fiber angle to increase), so the pressure decreases. It is not known how the fiber angle of the cuticle scales, nor how the material properties of the intact body wall scale (including the passive and active material properties of the muscle). Such information would help to determine the importance of elastic energy storage in explaining the observed patterns of segment shape change and internal pressure fluctuation.

In conclusion, soft-bodied organisms with hydrostatic skeletons are abundant and diverse and range in body mass over many orders of magnitude both within and among taxa. The present study used earthworms (Lumbricus terrestris) as an experimental system to begin an examination of how hydrostatic skeletons scale during ontogeny. Earthworms show geometric similarity and maintain static and dynamic stress similarity as they grow over several orders of magnitude in body mass. The hydrostatic skeletons of earthworms, therefore, differ fundamentally from the rigid lever-like skeletons of their terrestrial counterparts in their ability to grow isometrically while maintaining static stress similarity. Overall, the qualitatively distinct manner of mechanical function of hydrostatic skeletons and the relative unimportance of gravitational loads probably account to a great extent for the difference between the scaling of hydrostatic skeletons and terrestrial lever-like skeletons.

a y-intercept of log-transformed allometric power-law function

A cross-sectional area of a skeletal element bearing a force (m²)

A_f external area of application of force (m²)

A_i projected inside area of a hydrostat(m²) A_m cross-sectional area of muscle (m²)

b exponent of allometric power-law function

d diameter (m)

d₁₅ lateral diameter of segment 15 (m) d₅₀ lateral diameter of segment 50 (m) F force (N)

F_acc ground reaction force (N)

F_w force due to the weight of the organism (N)

g gravitational acceleration (m s⁻²)

L resting body length (m)

L_s resting segment length (m)

m_b body mass (g)

n number of segments

P internal pressure (N m⁻²)

P_C.M. internal pressure during circumferential muscle contraction (N m⁻²)

P_g gravitational pressure (N m⁻²)

P_L.M. internal pressure during longitudinal muscle contraction (N m⁻²)

P_m internal pressure due to muscle tension (N m⁻²)

P_R internal pressure at rest (N m⁻²)

r radius (m)

t body wall thickness (m)

t_15a annulus body wall thickness of segment 15 (m)

t_15s mid-segment body wall thickness of segment 15 (m)

t_50a annulus body wall thickness of segment 50 (m)

t_50s mid-segment body wall thickness of segment 50 (m)

U₁ rate of contraction of a muscle (m s⁻¹)

U₂ maximum velocity of a skeletal element (m s⁻¹)

z distance from upper surface of organism (m)

β regression slope corrected for attenuation

κ_xx reliability ratio

ρ fluid density (kg m⁻³)

σ_c circumferential stress (N m⁻²)

σ_l longitudinal stress (N m⁻²)

σ_m muscle stress (N m⁻²)

τ_max maximum shear stress in the wall of a cylinder (N m⁻²)

This research was supported by an 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 ONR Grant 444095-23068 to M. Koehl. I am especially grateful to M. Koehl for helpful discussion and advice, and to the members of the biomechanics group at the University of California at Berkeley for their insights and suggestions. I am grateful to M. Koehl, O. Ellers, M. Martinez and two anonymous referees for critical reading of this manuscript.