Symmorphosis and the insect respiratory system: allometric variation

In 1981, Taylor and Weibel, together with their co-workers, introduced the theory of symmorphosis. The principle was defined as the ‘state of structural design commensurate to functional needs resulting from regulated morphogenesis, whereby the formation of structural elements is regulated to satisfy but not exceed the requirements of the functional system’ (Taylor and Weibel, 1981). The assertion that each step in a biological system is quantitatively matched to the overall maximum functional task of the system is essentially an optimality theory, which was motivated by the belief that animals are built economically because maintaining superfluous structure is energetically expensive (Weibel et al., 1981a).

In a seminal series of papers, Taylor and Weibel tested their theory on the respiratory system of mammals. First, the functional capacity of the respiratory system was determined by measuring the maximum oxygen consumption rate of mammals during exercise of progressive increasing intensity (Seeherman et al., 1981). Next, a number of structural variables crucial in the delivery and consumption of oxygen during exercise were morphometrically quantified (Gehr et al., 1981; Hoppeler et al., 1981a; Hoppeler et al., 1981b; Mathieu et al., 1981; Weibel et al., 1981b). The effect of body mass on the maximum oxygen consumption rate of mammals and the amount of respiratory structure they possess was expressed as an allometric equation, y=aM ^b_b, where y is the variable of interest, a is the coefficient, M_b is body mass and b is the scaling exponent. Taylor and Weibel predicted that if their theory of symmorphosis holds for the mammalian respiratory system, then maximum oxygen uptake should scale with an exponent (b) that is statistically indistinguishable from the exponents derived for each of the structural variables (Taylor and Weibel, 1981). They found that maximum oxygen consumption rate scales with body mass with an exponent of 0.80 (Taylor et al., 1981), and that this is congruent with the allometric scaling of mitochondrial volume in the diaphragm and locomotory muscles, b=0.80 to 0.82 (Mathieu et al., 1981). It was also established that total capillary length in selected muscles scales roughly parallel with aerobic capacity, b=0.86 to 0.91, but because of scatter in the data it was not possible to determine whether the slopes were statistically similar (Hoppeler et al., 1981b; Weibel et al., 1981a). Interestingly, the anatomical diffusing capacity of the lung was found to scale with an exponent, b=0.99, significantly steeper than that derived for maximum oxygen consumption rate, which means that larger mammals appear to possess excess lung structure (Gehr et al., 1981). It was suggested that larger mammals may have a lower alveolar pressure head for diffusion and, therefore, require a greater pulmonary diffusing capacity to transfer oxygen from the lungs to the blood at the same rate as smaller mammals (Gehr et al., 1981). In contrast, the pulmonary anatomical diffusing capacity of birds appears to be symmorphic, both phylogenetically and ontogenetically (Canals et al., 2011; Runciman et al., 2005; Seymour et al., 2004; Seymour et al., 2008).

Since its conception, symmorphosis has received the support of a number of workers who view the optimal design idea as a useful null hypothesis of biological design on which to base analytical studies (Diamond, 1992; Suarez, 1996). Where these simple economy-based expectations break down, an opportunity arises to analyse the reasons for such a breakdown (Diamond, 1992). However, symmorphosis has also evoked criticism, especially from evolutionary biologists, who argue that natural selection leads to adequate or sufficient design but not optimal or symmorphic design (Dudley and Gans, 1991; Garland, 1998; Garland and Huey, 1987; Smith, 1978). They argue that symmorphosis is not an inevitable outcome because environmental unpredictability means that evolutionary selective forces vary too much in space and time to produce a structure that is functionally optimal (Dudley and Gans, 1991). It is suggested that symmorphic-like matching between structure and function may only ever be realised when the selective forces influencing design have either been static for a long time (Dudley and Gans, 1991) or when selective pressures have been directed towards extreme performance (Jones and Lindstedt, 1993). Criticism of symmorphosis is supported by a number of instances in which animals appear to be in possession of unnecessary structure. Not only does the mammalian lung appear to have an excess capacity (Canals et al., 2010; Gehr et al., 1981; Hsia et al., 1992; Karas et al., 1987), but excess structure may also exist at oxygen delivery steps further downstream. For example, maximum oxygen uptake in athletic humans, racehorses and mice increases under hyperoxic conditions (Gębczyński and Konarzewski, 2011; Jones and Lindstedt, 1993; Powers et al., 1989), which suggests that in some animals at least, the mitochondria have an aerobic reserve that is never fully exploited.

Whether there are physiological or adaptive explanations for instances where symmorphosis does not hold is open to debate, but first it is worth considering two other factors important in animal design. Firstly, it is likely that many physiological systems have a built-in safety margin, analogous to the safety margin engineers build in to bridges, buildings and elevators. If a structure has no safety margin, then it is at risk of breakage when its performance approaches maximum. For example, the cardiovascular system of racehorses has been artificially selected for high oxygen flux to the extent that the safety margin for pulmonary blood pressure has been significantly compromised, and when physically exerted some horses experience pulmonary bleeding (Pascoe et al., 1981). The second point worth considering is that some structures in the body are multifunctional and this can complicate the interpretation of results. For example, the design of the cardiovascular system could be geared towards oxygen delivery, or it could reflect some other physiological requirement, such as substrate delivery, lactate removal, carbon dioxide removal, pH regulation or heat dispersal (Dudley and Gans, 1991; Garland and Huey, 1987). Clearly, some systems offer a more suitable model on which to test for symmorphic design. The insect respiratory system appears to be a case in point.

In insects, oxygen delivery occurs along a series of steps (Hartung et al., 2004; Schmitz and Perry, 1999). First, atmospheric oxygen enters the air-filled tubes of the insect's tracheal system through small pores located along its lateral margins, called spiracles. It then moves in the gas phase through a branching system of successively smaller tracheae, before arriving at the terminal tracheoles. Here, it exits the tracheal system and diffuses across the tracheal cuticle and epidermal layers (Fig. 1) before diffusing into tissue where it is used by mitochondria to oxidise substrate to produce ATP for work. Carbon dioxide produced by the respiring cells does not take the exact same path as oxygen in reverse. Because of its greater solubility in tissue and haemolymph, some carbon dioxide may be temporarily retained in the tissue before diffusing into tracheae or passing directly through the integument (Bridges and Scheid, 1982). In some stick insects, for example, up to 25% of carbon dioxide is expelled directly across the exoskeleton (Wigglesworth, 1965). As such, a system capable of bringing an adequate supply of oxygen to the tissues should also suffice to remove carbon dioxide (Chapman, 1998; Schmitz and Perry, 1999).

The primary function of the insect tracheal system, therefore, is to deliver oxygen to the mitochondria, which makes it an ideal model on which to test for symmorphosis. However, there are other characteristics that make it appealing. Oxygen flux through the insect respiratory system is higher than in any other animal group (Sacktor, 1976; Suarez, 2000), and so the selective pressure for optimal design has likely been strong (Jones and Lindstedt, 1993). The high metabolic scope of insects also means that the oxygen requirements of the exercising muscle greatly exceeds metabolism in the remaining tissues, and this makes the work due to exercise easier to quantify. Furthermore, insect locomotory muscles are often easy to identify and thus analyse without including non-locomotory tissues. For example, hopping in locusts is performed by just two muscles contained within each metathoracic femur. Hopping is also entirely aerobic after 2 min of exercise (Kirkton et al., 2005), and for the purpose of an allometric study of symmorphosis, the fact that locust body mass increases more than 30-fold from first to fifth instar is also a key advantage.

The aim of this study was to test whether the respiratory system of the migratory locust, Locusta migratoria, is designed according to the economical principles of symmorphosis. To achieve this, allometric variation in the structural properties of the locust respiratory system in the hopping femurs was quantified and compared with the maximum functional rate at which oxygen flows through the system during intense hopping exercise in first, third and fifth instar juveniles. Earlier respirometry showed that the aerobic capacity of the hopping muscle () scales throughout juvenile locust development according to the allometric equation μmol O₂ h^–1 at 35°C (r²=0.98, N=145) (Snelling et al., 2011). If the theory of symmorphosis holds for the insect respiratory system, then similar scaling exponents should also exist for mitochondrial volume, mitochondrial inner membrane surface area, tracheal volume, tracheal surface area and tracheal diffusing capacity in the juvenile locust hopping muscle.

Gregarious-phase locusts Locusta migratoria (Linnaeus 1758) were reared at 33±1°C, under a 12 h:12 h light:dark cycle, with ad libitum access to seedling wheatgrass and wheat germ, as previously described (Snelling et al., 2011). The developmental stage of each locust was determined based on instar-specific differences in wing morphology. Newly moulted individuals were transferred into separate plastic terraria with other locusts of the same age. Morphometric quantification of the metathoracic femur (hopping leg femur) was conducted on locusts in their first, third and fifth instar life stages. Three insects were used from each life stage (N=9 individuals). Insects were measured 3–4 days post-moult to provide sufficient time for the exoskeleton to stiffen while minimising compression of the tracheal system due to growth (Greenlee and Harrison, 2004; Queathem, 1991). Locusts were fasted for 6–10 h prior to tissue fixation, consistent with earlier respirometry experiments (Snelling et al., 2011).

The procedure used to dissect and fix locust tissue was modified based on the methods of previous electron-microscopy studies of insects and spiders (Biserova and Pfluger, 2004; Hartung et al., 2004; Kohnert et al., 2004; Schmitz and Perry, 2002a; Schmitz and Perry, 2002b). The large extensor tibialis muscle and the smaller flexor tibialis muscle, which are used during hopping locomotion, constitute the entire muscle mass of the metathoracic femur (Bennet-Clark, 1975; Gabriel, 1985a; Gabriel, 1985b) and were identified from published illustrations (Albrecht, 1953).

Locusts were first cold anaesthetised for 20 min in a refrigerator at 4°C. They were then weighed to 0.1 mg on an analytical balance (AE163, Mettler, Greifensee, Switzerland). Both metathoracic femurs on each locust were then carefully removed with angled spring scissors. Each femur was weighed to 0.1 mg before being sliced along the transverse plane into pieces of equal length using a razor. Femurs from first instar locusts were sliced into three pieces, third instar femurs were sliced into six pieces, and fifth instar femurs were sliced into 12 pieces. Each piece of femur tissue was sequentially numbered proximally to distally and labelled left or right. The pieces were immediately immersed into a chemical fixative solution of 2.5% glutaraldehyde and 2% formaldehyde in 0.2 mol l^–1 phosphate buffer, pH 7.4, and left overnight in a refrigerator at 4°C. On the second day, each piece was given a series of buffer rinses. The first rinse was 20 min, followed by four 60 min rinses. This was followed by three 20 min distilled water rinses, and then the tissue was placed into a 1% aqueous solution of osmium tetroxide for secondary fixation and left overnight at room temperature. On the third day, each piece was given five 20 min rinses in distilled water and then placed into a 2% aqueous solution of uranyl acetate and left overnight at room temperature. On the fourth day, the pieces were given four 20 min rinses in distilled water and then dehydrated in ethyl alcohol in 10% incremental steps from 50 to 80%, each for 20 min. The tissue was then further dehydrated in consecutive 20 min immersion in 90% ethanol (2×), 100% ethanol (2×) and finally pure propylene oxide (2×). Following dehydration, the samples were incrementally infiltrated with embedding resin (Durcupan, Fluka, Buchs, Switzerland) at ratios of 3:1, 2:2 and 1:3 (propylene oxide to resin), respectively, each for a duration of 60 min, and then left overnight at room temperature in pure resin. On the fifth day, each piece was aligned longitudinally and in a proximal to distal orientation in individually labelled embedding moulds, where they were covered with pure embedding resin and left to polymerise in a 70°C oven for 48 h.

The procedure for sampling the embedded hopping muscle was essentially hierarchical [fig. 1.4 in Howard and Reed (Howard and Reed, 1998)]. Briefly, one metathoracic femur from each locust was randomly selected for light microscopy analysis, where 12 parallel, transverse and equidistant 1 μm thick sections were cut proximally to distally using 8 mm glass knives and an ultramicrotome (EM UC6, Leica Microsystems, Wetzlar, Germany). The sections were placed onto glass slides, stained with Toluidine Blue, rinsed with distilled water and air dried. Each section was viewed between ×4 and ×20 magnification under an optical microscope (BX51, Olympus, Hamburg, Germany) and photographed with a mounted digital colour camera (ColorView III, Soft Imaging System, Olympus) that generated 24 bit, 2576×1932 pixel resolution images. Each section image was imported into a computer graphics program (CorelDRAW 11, Corel Corp., Ottawa, ON, Canada) where a point grid test system was superimposed randomly over the image, which allowed transverse area and volume estimates using the following stereological equations (Cruz-Orive and Weibel, 1990; Howard and Reed, 1998; Mayhew, 1991).

While one metathoracic femur was randomly selected for light microscopy and Cavalieri volume calculation, the remaining femur was used for transmission electron microscopy analysis. The sampling scheme used for electron microscopy randomised section position and angle to produce vertical uniform random sections (Howard and Reed, 1998; Mayhew, 1991). Six equidistant 70 nm sections were cut proximally to distally along the leg using a diamond knife (Diatome, Biel, Switzerland) and ultramicrotome. For the first locust of each life stage, the first section was cut at a random angle between 0 and 18 deg relative to the vertical plane. For the second locust, the first section was cut at a random angle between 36 and 54 deg, and for the third locust, the first section was cut at a random angle between 72 and 90 deg. For the remaining five sections in each locust, 18 deg was added to the cutting angle with every successive section. This meant that for each locust femur, the full range of angles around the vertical plane was randomly sampled.

Sections were placed onto 3 mm copper mesh grids, stained with lead citrate and viewed with a transmission electron microscope (CM 100, Philips, Eindhoven, The Netherlands), where random 8 bit, 1280×1024 pixel resolution images were captured with a mounted digital camera (MegaView II, Soft Imaging System, Olympus) at appropriate magnifications (Table 1). To obtain unbiased sampling of the femur, it was necessary to weight the number of random images taken from different sections along the leg because the femur tapers significantly as it approaches the tibial joint. The mean transverse area distribution curve generated in Eqn 1 for each life stage was used to determine the extent that sampling should be weighted towards the larger proximal regions of the leg. Accordingly, the number of random images taken from each section along the femur was adjusted as a fraction of the total number of images taken per femur (Table 1). All images were exported to CorelDRAW for analyses, where a test system was randomly superimposed over the image, which allowed estimates to be made of volume density, surface area density and thickness using the following stereological equations (Cruz-Orive and Weibel, 1990; Howard and Reed, 1998; Mayhew, 1991) appropriate for vertical sections (Baddeley et al., 1986). The average time to process each locust from initial fixation to the final quantification of femur ultrastructure was ∼6 weeks. This significant investment in time restricted the study to nine individuals from three developmental stages.

The radial diffusing capacity for oxygen (i.e. radial conductance) out of the class I tracheae (tracheoles) and into the surrounding muscle was calculated for each locust according to Krogh's cylinder model for radial diffusion (Kreuzer, 1982; Krogh, 1919). Krogh's cylinder (Fig. 1) was adapted only for tracheoles because they are the most relevant in the supply of oxygen to the tissues (see Results).

Body mass increased 32-fold throughout juvenile development. First instars had a mean wet body mass of 0.019±0.001 g (N=3), which increased to 0.122±0.007 g by the third instar stage (N=3) and 0.611±0.010 g by the fifth instar stage (N=3). There was no significant difference between the body mass of locusts used in the present study and those used in earlier respirometry experiments (Snelling et al., 2011) for each life stage (Mann–Whitney U-test, P>0.05).

Muscle occupies approximately half the volume of the metathoracic femur (Fig. 2A,B), and muscle volume scales according to the allometric power equation V_muscle=4.3×10¹⁰M_b^1.05±0.08 μm³ (r²=0.99, N=9; Fig. 3B; supplementary material Table S1). Earlier respirometry experiments (Snelling et al., 2011) found that the maximum functional rate at which oxygen is consumed by jumping muscle during intense hopping exercise scales according to the equation μmol O₂ h^–1 at 35°C (r²=0.98, N=145). The exponent for is not significantly different to V_muscle (ANCOVA, F_1,150=0.43, P=0.51).

Mitochondria occupy approximately 3.5% of muscle cell volume regardless of life stage (Fig. 2C,D; Table 2). This is indicated in the allometric power equation for mitochondrial volume density, V_Vmito=0.032M_b^{–0.03±0.09} (r²=0.10, N=9; supplementary material Table S2).

The absolute volume of mitochondria in the hopping muscle is highly dependent on body mass and scales according to the equation V_mito=1.3×10⁹M_b^1.02±0.08 μm³ (r²=0.99, N=9; Fig. 3C; supplementary material Table S3). The exponent for V_mito is not significantly different to (ANCOVA, F_1,150=0.01, P=0.94).

The surface density of inner membrane within the mitochondria is similar for all life stages and averages approximately 33 μm² μm^–3 (Fig. 2E; Table 2). Inner mitochondrial membrane surface density scales according to the allometric power equation S_Vinner=31M_b^{–0.03±0.05} μm^–1 (r²=0.23, N=9; supplementary material Table S4).

The total surface area of inner mitochondrial membrane in the hopping muscle is highly dependent on body mass and scales according to the equation S_inner=4.2×10¹⁰M_b^0.99±0.10 μm² (r²=0.99, N=9; Fig. 3D). The exponent for S_inner is not significantly different to (ANCOVA, F_1,150=0.52, P=0.47).

The tracheal system occupies approximately 3% of the metathoracic femur volume. The volume of the tracheal system within both metathoracic femurs scales hypermetrically with body mass following the allometric power equation V_trach,sys=2.4×10⁹M_b^1.12±0.09 μm³ (r²=0.99, N=9; Fig. 3E; supplementary material Table S5), which has an exponent that is significantly greater than the exponent derived for (ANCOVA, F_1,150=4.44, P<0.05).

However, class IV tracheae and air sacs account for much of the tracheal system's volume: 68% in first instars, 71% in third instars and 78% in fifth instars (Table 3). The volume of class IV tracheae and air sacs increase disproportionately with body mass according to the power equation V_trachIV=1.9×10⁹M_b^1.16±0.16 μm³ (r²=0.98, N=9). If class IV tracheae and air sacs are excluded, then the volume of the tracheal system scales with a much lower exponent following the equation V_{trach,sys–trachIV}= 5.4×10⁸M_b^1.02±0.14 μm³ (r²=0.98, N=9), which has an exponent that is statistically indistinguishable from that derived for (ANCOVA, F_1,150=0.00, P=0.97).

Class II and III tracheae are relatively uncommon, representing just 5–10% of the tracheal system's volume despite their reasonably large size (Table 3).

Class I tracheae (tracheoles) contribute 10–15% of the tracheal system's volume (Table 3) and, because of their small size, are far more numerous than the other trachea size classes (Fig. 2B,D,F). The volume density of tracheoles in the muscle is largely invariant with body mass and scales according to the allometric equation V_VtrachI=0.0073M_b^0.06±0.24 (r²=0.04, N=9). The volume density of tracheole lumen in the muscle scales according to the equation V_VtrachI,lum=0.00051M_b^{–0.06±0.15} (r²=0.12, N=9). However, the absolute volume of tracheoles in the femurs is highly dependent on body mass and scales according to the equation V_trachI=3.1×10⁸M_b^1.11±0.23 μm³ (r²=0.95, N=9; Fig. 3F). The absolute volume of tracheole lumen in the femurs is less, and scales according to the equation V_trachI,lum=2.2×10⁷M_b^0.99±0.16 μm³ (r²=0.97, N=9; Fig. 3G). Neither the exponent for V_trachI (ANCOVA, F_1,150=2.87, P=0.09) nor that for V_trachI,lum (ANCOVA, F_1,150=0.35, P=0.56) are significantly different to .

The area of the tracheal system's outer epidermal surface in both femurs scales according to the allometric power equation S_{epi,trach,sys}=1.8×10⁹M_b^1.10±0.17 μm² (r²=0.97, N=9; supplementary material Table S5). The class I trachea (tracheole) outer epidermal surface area scales with juvenile body mass following the equation S_epiI=1.5×10⁹M_b^1.11±0.23 μm² (r²=0.95, N=9), which is not significantly different to the exponent derived for (ANCOVA, F_1,150=2.87, P=0.09).

The area of the tracheal system's inner cuticular surface in both femurs scales according to the allometric equation, S_{cut,trach,sys}=8.5×10⁸M_b^1.16±0.43 μm² (r²=0.85, N=9). The tracheole inner cuticle surface area scales with body mass following the equation S_cutI=2.0×10⁸M_b^0.99±0.16 μm² (r²=0.97, N=9; Fig. 3H), which is similar to the exponent derived for (ANCOVA, F_1,150=0.35, P=0.56).

The harmonic mean barrier thickness of the cuticular and epidermal layers increases with trachea size class (supplementary material Table S5). Class I tracheae (tracheoles) thus have the thinnest walls. The harmonic mean barrier thickness of the tracheole cuticle and epidermal layers does not vary with body mass. Tracheole harmonic mean barrier cuticle thickness scales according to the allometric equation τ_cutI=0.027M_b^{–0.01±0.05} μm (r²=0.05, N=9), and the harmonic mean barrier epidermal thickness scales according to the equation τ_epiI=0.19M_b^0.04±0.17 μm (r²=0.04, N=9).

The lateral diffusing capacity (lateral conductance) of the entire tracheal system in the femurs scales isometrically with body mass following the equation G_L,trach,sys=23M_b^1.00±0.21 μmol O₂ kPa^–1 h^–1 at 35°C (r²=0.95, N=9; supplementary material Table S6), which is not significantly different to the exponent derived for (ANCOVA, F_1,150=0.21, P=0.64).

Class I tracheae (tracheoles) provide ∼90% of the lateral diffusing capacity of the entire tracheal system in the femurs (Table 4). The lateral diffusing capacity of tracheoles scales according to the allometric equation G_L,trachI=20M_b^0.99±0.17 μmol O₂ kPa^–1 h^–1 at 35°C (r²=0.96, N=9; Fig. 3I), which is also not significantly different to the exponent derived for (ANCOVA, F_1,150=0.40, P=0.53).

The mean transverse area of muscle serviced by each tracheole is 440 μm² (radius=12 μm) and scales independently with body mass according to the allometric relationship T_muscleI=482M_b^0.06±0.15 μm² (r²=0.12, N=9).

The radial diffusing capacity (radial conductance) for oxygen through the tracheole cuticle, epidermis and surrounding muscle scales with juvenile body mass according to the allometric equation G_R,trachI=5.8M_b^0.99±0.18 μmol O₂ kPa^–1 h^–1 at 35°C (r²=0.96, N=9; Fig. 3J; supplementary material Table S6), which is statistically similar to the exponent derived for (ANCOVA, F_1,150=0.55, P=0.46).

Respirometry conducted prior to the present study revealed that the maximum functional rate at which oxygen is delivered and consumed by the juvenile hopping muscle during strenuous jumping exercise, , scales with body mass with an exponent of 1.02±0.02 (Snelling et al., 2011). According to the principles of symmorphosis, the structures involved in the delivery (tracheal system) and consumption (mitochondria) of oxygen should scale congruently with maximum oxygen consumption rate (Weibel et al., 1991).

The volume of hopping muscle contained within the metathoracic femurs (Fig. 2A,B) scales with an exponent of 1.05 (Fig. 3B). Contained within the muscle fibres are mitochondria (Fig. 2C,D) that occupy approximately 3.5% of muscle volume in all locusts (Table 2), which agrees well with the results of another study that estimated a mitochondrial density of approximately 2–4% in the hopping muscle of American locusts, Schistocerca americana (Hartung et al., 2004). However, it is considerably less than the 20–50% mitochondrial density estimated for insect flight muscle, which is no doubt related to the much higher energetic requirements for flight compared with terrestrial locomotion (Mizisin and Ready, 1986; Smith, 1963).

The volume of mitochondria in the hopping muscle scales with an exponent of 1.02 (Fig. 3C) and is parallel with the scaling of the hopping muscle's aerobic capacity. The congruent scaling of structure and function suggests that no more mitochondria exist than are necessary to meet the maximum energy requirements of the exercising muscle, which is indicative of symmorphic design. Matching at this step of the oxygen cascade has also been shown in the locomotory and diaphragm muscles of mammals (Mathieu et al., 1981; Schwerzmann et al., 1989; Weibel et al., 2004).

Within the mitochondria, the respiratory enzymes for oxidative phosphorylation are located in the inner membrane (Fig. 2E), and so it is reasonable to argue that quantifying the inner membrane surface area is a more relevant test for symmorphosis. Mitochondria in locust hopping muscle have a mean inner membrane surface density of approximately 33 μm² μm^–3 (Table 2). This is similar to cat muscle mitochondria, which have an inner membrane surface density of approximately 35 μm² μm^–3 (Schwerzmann et al., 1989), but is much less than the flight muscle of flies (Smith, 1963) and hummingbirds (Suarez et al., 1991), which are in the vicinity of 50–60 μm² μm^–3. It is speculated that in these animals, the inner membrane surface density is at the upper limit, beyond which there would be insufficient room in the mitochondrial matrix for Krebs cycle enzymes to function properly (Suarez, 1996).

The mitochondrial inner membrane surface density in hopping muscle decreases very slightly with age (Table 2), which probably reflects the gradual increase in mitochondrion size that occurs during development rather than a decrease in the density at which cristae are packed. The absolute surface area of mitochondrial inner membrane therefore scales with an exponent of 0.99 (Fig. 3D), which is statistically similar to the exponent derived for the hopping muscle's aerobic capacity. Therefore, at this final step of the oxygen cascade, the principles of symmorphosis are evident.

The volume of the entire tracheal system within the metathoracic femur scales with an exponent of 1.12 (Fig. 3E), which is significantly greater than the scaling of the hopping muscle's maximum oxygen consumption rate. This suggests that larger locusts may have more tracheae than they require, which runs counter to the principles of economic design implicit with the symmorphosis hypothesis. However, it is worth examining each trachea size class separately because different sized tracheae may perform different functions.

More than half the tracheal volume can be attributed to large class IV tracheae and air sacs (Table 3), which scale with a rather steep exponent of 1.16. This means that large locusts have relatively more large-sized tracheae. However, the harmonic mean barrier thickness of class IV tracheae is very thick, and so although they represent more than half the total tracheal volume, they provide less than 5% of the overall lateral diffusing capacity (Table 4). The large size and poor lateral conductance of class IV tracheae suggest that their main function is to ventilate the tracheal system (Maina, 1989; Weis-Fogh, 1967; Westneat et al., 2003). If this is correct, then the disproportionately greater volume of class IV tracheae in older locusts could be necessary to overcome the physiological diffusion limitations associated with long tracheal tubes in larger insects (Greenlee et al., 2009; Harrison et al., 2005; Kaiser et al., 2007). The idea that large insects are diffusion limited is central to the hypothesis that insect gigantism during the late Palaeozoic was facilitated by hyperoxic atmospheres that occurred around the same time (Dudley, 1998; Graham et al., 1995).

Class II and III tracheae each represent just ∼3% of the tracheal system's volume and total surface area in the femur (Table 3). Compared with larger class IV tracheae, their walls are thinner and so they probably allow a small amount of oxygen to diffuse across, but because they are so uncommon, they account for less than 3% of the tracheal system's overall lateral diffusing capacity (Table 4). Therefore, it seems that the primary function of class II and III tracheae is to serve as an intermediary between class IV and class I tracheae.

Small class I tracheae (tracheoles) represent the terminal endings of the tracheal system and contribute 13% to overall tracheal volume and less than 1% to muscle volume in the femur (Table 3). Tracheole volume scales with an exponent of 1.11 (Fig. 3F), which is higher than the 1.02 derived for the hopping muscle's aerobic capacity, but the difference is not significant. However, oxygen is delivered along the tracheoles in the gas phase, and so it is more useful to analyse the lumen volume only, which excludes the cuticular and epidermal components that are important only in the across-wall diffusion of oxygen. The lumen volume of tracheoles scales with an exponent of 0.99 (Fig. 3G) and this is well matched to the aerobic capacity of the juvenile hopping muscle. Thus, although the volume of the entire tracheal system is not symmorphic, the functional part is.

To reach the exercising muscle, oxygen must first diffuse across the tracheole inner cuticle layer followed by the outer epidermal layer. The tracheole outer epidermal layer has a surface density of approximately 4 μm² μm^–3, and its total surface area scales with body mass with an exponent of 1.11, which is somewhat higher than the exponent derived for the hopping muscle's maximum oxygen consumption rate. However, given that the outer epidermal surface covers a greater area than the inner cuticle surface, it would seem that the inner cuticle layer is far more important in terms of limiting oxygen flow out of the tracheoles (Fig. 2F; Table 3). The tracheole inner cuticle layer has a surface density of approximately 8 μm² μm^–3, and its total surface area scales with body mass with an exponent of 0.99 (Fig. 3H), which is congruent with the hopping muscle's aerobic capacity as predicted by symmorphosis.

The lateral diffusing capacity of the entire tracheal system within the femurs scales isometrically with body mass and approximately 90% of the system's lateral conductance for oxygen can be attributed to the tracheoles (Table 4), which are intimately associated with the muscle (Fig. 2B,D,F). Therefore, tracheoles represent the primary site for oxygen transfer from the tracheal system to the muscle, which is in agreement with the results of other morphometric studies on insects (Hartung et al., 2004; Schmitz and Perry, 1999). The harmonic mean barrier thickness of the tracheole cuticle and epidermal layers does not vary much throughout development, and so the tracheole lateral diffusing capacity scales with an exponent of 0.99 (Fig. 3I), following a slope similar to that of the inner cuticle surface area. The exponent for tracheole lateral diffusing capacity matches the scaling of the hopping muscle's maximum oxygen consumption rate, as predicted by symmorphosis.

However, the congruent scaling between lateral diffusing capacity and the hopping muscle's aerobic limits is a little surprising because the ontogenetic growth of the hopping muscle is associated with an observed increase in the size of muscle cells [also observed by Hartung et al. (Hartung et al., 2004)], which would lead to longer average diffusion distances for oxygen through the muscle. In response to the longer diffusion distance, it is reasonable to suspect that larger locusts may have a disproportionate investment in the abundance of tracheoles as a means to compensate for this. Instead, larger locusts appear to overcome this potential diffusion problem in part by investing more in large class IV tracheae (Table 3), which function to ventilate the tracheal system and maintain a high pressure head for oxygen diffusion (Maina, 1989; Weis-Fogh, 1967; Westneat et al., 2003), and in part by increasing the number of tracheoles that penetrate the muscle cell, such that they become functionally intracellular and, therefore, in closer proximity to mitochondria deep within the muscle fibre (Fig. 2D). Small first instars lack these intracellular tracheoles; however, in large fifth instars, 5% of tracheoles are intracellular. The occurrence of tracheoles that penetrate the surface of other cells has been reported in a number of studies, although they appear to be particularly prevalent in large and highly aerobic flight muscle fibres (Whitten, 1972; Wigglesworth, 1983).

The observation that some tracheoles are functionally intracellular leads to the realisation that the lateral diffusion of oxygen may not be the most appropriate model to apply to tracheoles because it does not account for the extra distance oxygen must travel as it diffuses through the surrounding muscle, or the inevitable dissipation of oxygen molecules as they diffuse further from the tracheole source. In this respect, the flow of oxygen out of tracheoles and into the surrounding muscle in insects is analogous to the flow of oxygen out of capillaries and into the surrounding tissue in vertebrate animals. Therefore, Krogh's cylinder for radial diffusion should offer a more appropriate model on which to estimate the true anatomical conductance for oxygen out of the tracheoles and into the hopping muscle. The results of the present study show that the mean transverse area of muscle serviced by each tracheole is approximately 440 μm² and this does not vary significantly with increasing body mass. Consequently, the mean maximum distance oxygen must diffuse from a tracheole into the muscle fibre is ∼12 μm in locusts at all stages of development. This is comparable to a maximum oxygen diffusion distance of 8 μm in the muscle capillary network of the 2 g Etruscan shrew (Hoppeler et al., 1981b). From the estimate of maximum diffusion distance, it was calculated that the tracheole radial diffusing capacity scales with an exponent of 0.99 (Fig. 3J), which is congruent with both the tracheole lateral diffusing capacity and the aerobic capacity of the hopping muscle. The matching between anatomical tracheole diffusing capacity and the aerobic capacity of the hopping muscle is evidence of symmorphosis in the insect tracheal system, and complements the results of an earlier ontogenetic study on American locusts, which showed that the convective capacity of large tracheae located between the spiracles and the digestive tract matches the metabolic rate of locusts shortly after feeding (Harrison et al., 2005).

The average oxygen partial pressure in the terminal tracheoles required to overcome the diffusion barrier and meet the oxygen needs of the hopping muscle during heavy exercise is approximately 17 kPa at 35°C or, expressed as an allometric equation, P_O2,trachI=19M_b^0.05±0.15, kPa at 35°C (r²=0.08, N=9). It is not possible to measure directly the partial pressure of oxygen in the terminal tracheoles given their small size (<2 μm diameter). However, in resting American locusts, direct measurements in the upper regions of the tracheal system reveal that the partial pressure of oxygen is 19 kPa (Gulinson and Harrison, 1996). Given that oxygen is delivered along the tracheal system in the gas phase, it is reasonable to suspect that there may not be a large pressure drop in oxygen between the upper regions of the tracheal system measured by Gulinson and Harrison (Gulinson and Harrison, 1996) and the terminal tracheoles estimated in the present study. Therefore, our calculated estimate of approximately 17 kPa in the terminal tracheoles seems reasonable.

Along each step of the oxygen cascade there is matching between the maximum functional rate at which oxygen flows through the respiratory system and the quantitative investment made into the structures of the respiratory system. This provides strong evidence that the insect respiratory system is designed according to the economic principles of symmorphosis. However, it should be recognised that the variation in the scaling exponents for the tracheal system might reduce the power of an ANCOVA to distinguish differences with the scaling of the hopping muscle's aerobic capacity. This variation exists despite the analysis of 108 light micrographs and 1326 electron micrographs from nine individuals spanning a 32-fold range in body mass.

Some of the variation in the tracheal exponents could arise because the allometric equation may not accurately reflect changes in tracheal dimensions that occur during ontogeny. While muscle volume increases continually throughout development, the tracheal system undergoes significant expansion only at moulting (Ryerse and Locke, 1978; Wigglesworth, 1981). Therefore, in reality the dimensions of the tracheal system during ontogeny would likely resemble a series of five instar steps, rather than a straight line, and this has probably lead to some of the variation. Taking this into account, it is plausible that the reason the third instar locusts consistently lie below the tracheole regression lines (Fig. 3F–J) could be because they were fixed late in their life stage, when the ratio between muscle and tracheole volume is at its widest. Either way, it is unlikely that the third instars bear a strong influence on the final scaling exponent, given that the allometric slope tends to be more affected by data at the upper and lower regions of the analysed body mass range.

The other point worth considering is that in the adult locust, the flight muscles become functional and the insect is capable of oxygen consumption rates far in excess of those attained during terrestrial locomotion. It is reasonable to argue that symmorphosis might be most evident in respiratory systems geared toward extreme performance, such as insect flight. To this end, we have tested whether the vastly different aerobic metabolic rates attained during hopping and flight in the adult locust is matched by a proportional difference in the amount of respiratory structure in the hopping and flight muscles. The results of this study will be published at a later date.

 
• G_L

  lateral diffusing capacity for O₂ (i.e. lateral conductance)

 
• G_R

  radial diffusing capacity for O₂ (i.e. radial conductance)

 
• K

  Krogh diffusion coefficient (corrected to 35°C using Q₁₀=1.1)

 
• M_b

  body mass

 
• 

  maximum oxygen consumption rate of juvenile locust hopping muscles

 
• r_cut

  mean radius to inner surface of tracheole cuticle layer

 
• R_cut

  mean radius to outer surface of tracheole cuticle layer

 
• R_epi

  mean radius to outer surface of tracheole epidermal layer

 
• R_muscle

  mean radius of muscle serviced by each tracheole

 
• S

  surface area

 
• S_V

  surface area density

 
• T

  transverse area

 
• V

  volume

 
• V_V

  volume density

 
• τ

  harmonic mean barrier thickness

Mr Kerry Gascoigne and Dr Mike Teo of Flinders Microscopy, and Ms Ruth Williams and Ms Lyn Waterhouse of Adelaide Microscopy shared their expertise in the field of electron microscopy, which was greatly appreciated. The authors also thank Prof. Stephen Simpson and Mr Tim Dodgson from the University of Sydney for technical advice and providing the founding locust colony. We also acknowledge two anonymous reviewers who improved an earlier version of this manuscript.