The influence of oxygen and high-energy phosphate diffusion on metabolic scaling in three species of tail-flipping crustaceans

Predator–prey interactions result in strong selection for efficient escape responses in prey, and in some crustaceans a tail-flip response provides this vital function. Tail-flipping consists of fast backward propulsion resulting from the contraction of the abdominal musculature in response to a predator attack or other external stimuli. The anaerobic fibers that power tail-flipping in some species of crustaceans are often among the largest cells in the animal kingdom. Mammalian muscle cells typically range from approximately 10 to much less than 100 μm in diameter(Russell et al., 2000). This fundamental design constraint yields a high fiber-surface-area to volume ratio(SA:V) and short intracellular diffusion distances. Exceeding this maximum size might therefore compromise aerobic metabolism, which relies on oxygen flux from the blood to the mitochondria and ATP-equivalent diffusion within the cytoplasm (Mainwood and Rakusan,1982; Meyer, 1988; Hubley et al., 1997; Boyle et al., 2003; Johnson et al., 2004; Kinsey et al., 2005; Hardy et al., 2006). However,some crustacean anaerobic fibers can be hundreds of microns in diameter(Hoyle et al., 1973; Kinsey and Ellington, 1996; Boyle et al., 2003). Furthermore, muscle growth in crustaceans largely occurs hypertrophically– that is, muscle mass increases owing to an increase in fiber diameter and length, whereas fiber number remains nearly constant(Bittner and Traut, 1978; Boyle et al., 2003). As some species of crustaceans undergo an extreme increase in body mass during development, there is a parallel increase in muscle fiber size. Thus,crustacean muscle fibers from juveniles fall within the fiber size range typical of many animals, but, as the animals grow, the fibers can greatly exceed the usual threshold on cellular dimensions, while still preserving function (Boyle et al., 2003; Johnson et al., 2004).

During tail-flipping, energy requirements exceed aerobic capacity and contraction is anaerobic and therefore is not constrained by diffusion of O₂ to mitochondria or ATP from mitochondria(England and Baldwin, 1983). Following bursts of tail-flipping, recovery must occur before the animal can generate another round of high-force contractions. Metabolism during an anaerobic burst–escape response in crustacean muscle follows the same pattern as in vertebrates. Contraction is initially powered by the hydrolysis of the phosphagen arginine phosphate (AP), which is the crustacean analog of phosphocreatine (PCr) in vertebrates. Once AP pools are nearly depleted, ATP for additional contractions is supplied by anaerobic glycogenolysis, which is reflected by the accumulation of lactate and depletion of glycogen(England and Baldwin, 1983; Booth and McMahon, 1985; Head and Baldwin, 1986; Milligan et al., 1989; Morris and Adamczewska, 2002). These glycogenolytically powered contractions are slower and less forceful than those powered by phosphagen hydrolysis(England and Baldwin, 1983; Head and Baldwin, 1986; Baldwin et al., 1999; Boyle et al., 2003).

In contrast to burst contraction, metabolic recovery followingcontraction in crustaceans does not always follow the vertebrate paradigm. Vertebrates rely exclusively on aerobic metabolism to power resynthesis of creatine phosphate, and lactate does not accumulate following contraction(Kushmerick, 1983; Meyer, 1988; Curtin et al., 1997), whereas it is widely known that crustaceans often produce lactate after contraction(i.e. Head and Baldwin, 1986; Henry et al., 1994). In addition, Johnson and colleagues (Johnson et al., 2004) found that post-contractile lactate accumulation was fiber size dependent in blue crabs, in which large fibers produced significant amounts of lactate after contraction, whereas small crabs had no significant increase. These findings are consistent with those of Boyle and colleagues(Boyle et al., 2003), who reported significant post-contractile depletion of glycogen in aerobic fibers from large, but not small, crabs. Presumably, adult crabs are exploiting anaerobic metabolism to accelerate certain key phases of recovery such as AP resynthesis, the rate of which would otherwise be restricted by large fiber size (Kinsey and Moerland,2002; Boyle et al.,2003; Johnson et al.,2004; Kinsey et al.,2005). It also has been suggested that, as animal size increases,the negative allometry of aerobic metabolism would place heightened demands on anaerobic glycogenolysis during recovery(Baldwin et al., 1999). As anaerobic metabolism utilizes endogenous substrates and does not require oxygen, the scope of metabolic energy expenditure becomes limited only by maximal enzyme activity (Wells et al.,2001). Thus, anaerobic glycogenolysis is not limited by diffusive constraints, making it an effective means of accelerating AP resynthesis in large fibers. However, reliance on anaerobic glycogen to speed recovery will put the animal further in oxygen debt, and complete recovery must ultimately be aerobic.

The fiber-size dependence of metabolic recovery described above is consistent with intracellular diffusion limitation. However, previous studies using reaction–diffusion models suggest that the intracellular diffusive flux of high-energy phosphate (HEP) molecules and associated metabolites (ATP,ADP, P_i, AP/PCr and arginine/creatine) does not lead to sizable intracellular concentration gradients in the very large anaerobic fibers of crustacean or fish muscle, despite the fact that diffusion might occur over hundreds of microns (Kinsey et al., 2005; Hardy et al.,2006; Nyack et al.,2007). It should be noted, however, that many fibers in adult animals appear to be as large as possible without encountering severe diffusion limitations (Kinsey at al.,2007). The absence of strong concentration gradients in HEP molecules suggests that low fiber SA:V and limited O₂ flux might be a more important factor to consider when examining the rate of recovery in large fibers. Evidence supporting this argument includes previously observed shifts in the distribution of mitochondria towards the periphery of the fiber during growth of crustacean and fish white muscle, which might help counteract the low SA:V in the large fibers of adults(Boyle et al., 2003; Nyack et al., 2007). In addition, the open circulatory system present in crustaceans is associated with relatively low extracellular P_O2(Forgue et al., 2001), which might lead to rates of O₂ delivery that are insufficient to fuel high aerobic fluxes (Raffin et al.,1988; Mente et al.,2003).

The results described above imply an effect of muscle fiber size on aerobic metabolism in the blue crab; however, it is not known whether these effects constitute a fundamental diffusion constraint on cell design that can be broadly observed. Furthermore, it is not clear whether concentration gradients within the fiber actually lead to an alteration of aerobic metabolic flux. In the current study, the objectives were to examine the relationship between the scaling with body mass of whole-animal metabolic rate (rate of O₂consumption, or V̇_O2), as well as the scaling of muscle aerobic capacity (citrate synthase activity) and a muscle metabolic process (post-contractile AP resynthesis) in three groups of tail-flipping crustaceans: the grass shrimp (Palaemonetes pugio),brown and pink shrimp (Penaeus spp.) and the spiny lobster(Panulirus argus). These species differ dramatically in body mass range during development, which facilitates both an intraspecific and interspecific analysis of scaling. We analyzed the observedpost-contractile muscle recovery rates by developing a mathematical reaction–diffusion model that accounts for both O₂ and HEP diffusive flux and allows the evaluation of the influence of diffusion on aerobic flux. This is an advance over previous models that considered only the magnitude of the concentration gradients. This approach allowed us to determine the extent to which diffusion limits aerobic metabolism in muscle and potentially alters scaling relationships.

The current study used a small (juvenile) and large (adult) size class for each species. Grass shrimp (P. pugio Holthius) were collected from River Road Park in Wilmington, NC, USA, and brown shrimp (P. aztecusIves) and pink shrimp (P. duorarum Burkenroad) were collected by trawl offshore of Masonboro Inlet, NC, USA or ordered from Gulf Specimens(Panacea, FL, USA). Brown shrimp and pink shrimp are morphologically and behaviorally similar, and we found no differences between species in any of the measurements that we made. Therefore, both species were pooled as Penaeus spp. Juvenile P. argus Latreille were ordered from the Keys Marine Laboratory, FL, USA, and adult P. argus were collected by SCUBA diving 40 miles offshore of Wilmington, NC, USA. Water temperatures during collection for all species were 20–25°C as animals from NC were collected during the summer months. All animals were maintained in tanks containing filtered sea water that was constantly aerated. Smaller animals were maintained in glass aquaria at 20–22°C and a 12 h:12 h L:D photoperiod. Larger animals were maintained in 635 litre fiber-glass aquaria under the same conditions. Temperature was monitored daily, and water quality (nitrates, nitrite, pH and ammonia) was monitored weekly using a Salt Water Master Kit (Aquarium Pharmaceuticals, Chalfont, PA,USA). All animals were held for at least 1 week before experimentation. They were fed every other day, although feeding was stopped 24 h before any of the oxygen consumption trials to prevent spuriously high metabolic rates due to digestive activities.

Closed-chamber respirometry was used to measure the standard and post-contractile mass-specific rate of oxygen consumption(V̇_O2). V̇_O2 was measured using a YSI Model 5300 dissolved oxygen meter (Yellow Springs, OH, USA), using chambers with capacities of 20, 425, 800, 4000, 8000 and 26,000 ml. Before running an oxygen-consumption trial, the dissolved-oxygen electrode was calibrated with air-saturated sea water to ensure 100% oxygen saturation. Animals were placed in the chamber and were left to acclimatize for an hour before measurements. All trials were conducted during daylight hours, and chambers were covered to avoid visual disturbances that might cause increases in standard metabolic rate. During the acclimatization time, the water was constantly air saturated to prevent any possible oxygen depletion. After acclimatization, the chamber was submerged in salt water to eliminate bubbles inside the chamber, and it was then promptly sealed. V̇_O2 was continuously recorded using a Servogor 102 chart recorder until O₂in the chamber dropped to 70% air saturation. Following the measurement of standard metabolic rates, the animals were provoked to tail-flip a predetermined number of times. The number of tail-flips that led to exhaustion in each species was measured in pilot studies, and the numbers of tail-flips used for measurements of V̇_O2 were half of those previously measured as exhaustive work. Following the exercise protocol,the chamber was again sealed, and post-contractile V̇_O2 was measured. Animals were allowed to consume 30% of the oxygen in the chamber before measurements were stopped.

Tail muscle was excised, mounted using optimal cutting temperature (O.C.T.)compound (Sakura Finetek, Torrance, CA, USA) and allowed to equilibrate to–19°C in a Reichert–Jung/Leica cryocut 1800 (Depew, NY, USA)before sectioning. Sections were cut at 20 μm using the cryocut microtome,picked up on slides and air-dried at room temperature. Sections were incubated for 30 min in a 25 mg/ml solution of the lectin wheat-germ agglutinin, which binds to sugars on the surface of endothelial cells in vascular pathways. Slides were then rinsed in the appropriate saline solution (for lobsters, a solution containing 452 mmol l^–1 NaCl, 15 mmol l^–1 KCl, 18.9 mmol l^–1 CaCl₂, 4 mmol l^–1 MgCl₂, 2.8 mmol l^–1MgSO₄ pH 7.4 and, for shrimps, a solution containing 525 mmol l^–1 NaCl, 13.3 mmol l^–1 KCl, 12.4 mmol l^–1 CaCl₂, 24.8 mmoll^–1MgCl₂ pH7.4) for 60 min. Stained slides were examined with an Olympus BX-60 microscope (Center Valley, PA, USA), and images were recorded with a SPOT RT-KE camera. Polygons of fiber diameters were traced using Adobe Photoshop (version 7.0), and Image Pro Plus (version 6.0) was used to analyze fiber diameters. The average diameter of the polygon traces through the centroid was calculated in two-degree increments around the circumference of the cell.

Citrate synthase (CS) activity assays were conducted based on the methods of Walsh and Henry (Walsh and Henry,1990). Abdominal muscle was excised and freeze-clamped in liquid nitrogen and stored in plastic vials at –80°C until use. Tissue was extracted in 5–20 volumes of enzyme extraction buffer containing 50 mmol l^–1 Tris, 1 mmol l^–1 EDTA, 2 mmol l^–1 MgCl₂, 2 mmol l^–1 DTT at a pH of 7.6 and then sonicated on ice at 10 W in three bursts of 5 s each. Samples were centrifuged for 20 min at 16,000 g, and the supernatant was combined with 1 mmol l^–1 5,5-dithio-bis (2-nitobenzoic acid), 0.3 mmol l^–1 acetyl coenzyme A and water. Absorbance at a wavelength of 412 nm was measured in an Ultrospec 4000 spectrophotometer(Amersham-Pharmacia Biotech, Buckinghamshire, UK) at 25°C until the absorbance stabilized. The absorbance change during this period was slight and typically stabilized in 2–3 min. The reaction was initiated by the addition of 0.5 mmol l^–1 oxaloacetate, and the enzyme activity was determined from the initial slope of the absorbance change.

During a burst exercise–recovery cycle, there is a reciprocal change in AP and inorganic phosphate (P_i) that results from the stoichiometric coupling of cellular ATPases and the arginine kinase reaction. Contraction results in a rapid depletion of AP, and corresponding increase in P_i, which is followed by a slow recovery to pre-contractile levels, where the initial phase of recovery is the most rapid. To characterize the maximal rate of AP recovery, animals were exercised following the above protocol, and, after a pre-determined recovery period, the abdomen was rapidly removed by a swift cut between the abdomen and the cephalothorax. The first segment of the abdominal musculature was removed from the animal. The muscle was immediately freeze-clamped in liquid nitrogen. Tissues were immediately homogenized in a 5–60-fold dilution of chilled 7% perchloric acid with 1 mmol l^–1 EDTA and then centrifuged at 16,000 g for 30 min at 4°C. The supernatant pH was neutralized with 3 mol/l potassium bicarbonate in 50 mmol l^–1PIPES, stored on ice for 10 min and centrifuged at 16,000 gfor 15 min at 4°C. The supernatant was immediately analyzed by ³¹P nuclear magnetic resonance (NMR) spectroscopy. NMR spectra were collected at 162 MHz on a Bruker 400 DMX spectrometer (Billerica, MA, USA) to determine the relative concentrations of AP, ATP and P_i. Spectra were collected using a 90° excitation pulse and a relaxation delay of 12 s, which ensured that the phosphorus nuclei were fully relaxed and peak integrals for the metabolites were proportional to their relative concentrations. The area under each peak was integrated using Xwin-NMR software to yield the relative concentrations of each metabolite, and these values were converted to concentration by assuming a total HEP concentration of 50 mmol l^–1, which is characteristic of crustacean white muscle (see Kinsey et al.,2005). The initial slope of AP recovery was essentially linear with time, and so we used linear regression to determine the maximal rate of AP resynthesis following contraction.

Muscle lactate was measured immediately after exercise and following 15 min of recovery to estimate the fiber size dependence of post-contractile reliance on anaerobic metabolism. Frozen tissue samples were homogenized in 5–60-fold dilutions of chilled 7% perchloric acid with 1 mmol l^–1 EDTA, and then centrifuged at 4°C at 16,000g for 30 min. The supernatant was neutralized using 3 mol/l potassium bicarbonate in 50 mmol l^–1 PIPES and centrifuged at 4°C at 16,000g for 15 min. The resulting supernatant was stored at –80°C until use. The concentration of l-lactate in the tail musculature of crustaceans was spectrophotometrically assayed following the procedures of Lowry and Passonneau (Lowry and Passonneau,1972), as modified by Kinsey and Ellington(Kinsey and Ellington, 1996). A buffer containing 300 mmol l^–1 hydrazine hydrate, 12 mmol l^–1 EDTA and 4 mmol l^–1 NAD⁺ at pH 9.0 was mixed with the tissue extract in a 0.5 ml cuvette, and absorbance was monitored at a wavelength of 340 nm to obtain a stable baseline. The reaction was initiated by the addition of 18.5 units of l-lactate dehydrogenase, and the change in absorbance was measured. The concentration in the sample was calculated by assuming that 1 g of muscle tissue has 0.75 ml of intracellular water (Milligan et al.,1989).

The influence of body mass on specific V̇_O2, CS activity and AP recovery were analyzed using linear regression analysis, and analysis of covariance (ANCOVA) was used to compare scaling relationships. Fiber diameter was compared between each size class within a species using Student's t-tests. Student's t-tests were also used to compare lactate concentration immediately after exercise with the concentration 15 min post exercise. Results were considered significant if P<0.05.

The concentration of oxygen at the source is either 2.50 μmol l^–1 (low), 7.85 μmol l^–1 (intermediate)or 35.33 μmol l^–1 (high), where the latter two cases are based on experimental measurements in crustaceans(Forgue et al., 2001).

The first set of calculations was determined using Eqn 6 to find the concentrations in the absence of diffusion for various values of k₁ and k₂. The resulting rates were determined by Eqn 8. In the absence of diffusion, it was found that multiple values of k₁ and k₂ could satisfy the reaction rate; however, there were minimal values of k₁ below which the rate could not be attained. Similar analysis was conducted in the case with diffusion to determine the rate using the numerical solution of Eqn 3 and evaluating the rate with Eqn 9. As in the no-diffusion case, it was found that a range of combinations of k₁ and k₂ can give the same reaction rate; therefore, another set of calculations for the cases in the presence of diffusion was performed to determine the value of k₂ for various fixed values of k₁ that would match the experimentally determined reaction rate. In this set of computations, the value of k₁ was set at fixed values of the smallest value that would satisfy the rate. For each k₁ value, a root-finding method was used to determine the value of k₂that would give the desired experimental rate. The average concentrations of oxygen and ATP were determined in each of these cases (with fixed rate) by using Eqn 5. As a range of k₁ and k₂ can satisfy the rate, it is important to note that the average ATP and oxygen concentrations change– that is, the average ATP concentration – drops with increasing k₁.

Grass shrimps ranged in body mass from 0.045 to 0.624 g, with mean body masses for the small and large size classes of 0.11±0.006 and 0.40±0.017 g, respectively; penaeid shrimps ranged from 0.96 to 25.3g,with means for the small and large size classes of 3.7±0.223 and 10.5±0.466 g, respectively; and lobsters ranged from 70.4 to 3105g,with means for the small and large size classes of 151.7±5.33 and 2950±35.6 g, respectively. This corresponded to a 69,000-fold body mass range across all species and a nearly 27,000-fold range in mean body mass across all species size classes.

The allometric scaling equation, y=aM^b, was used to evaluate aerobic metabolism as a function of body mass, where y is the measured physiological property, M is body mass, a is a constant, and b is the scaling exponent(Schmidt-Nielsen, 1984). Mass-specific metabolic rate was lower in larger animals, as expected, and post-exercise V̇_O2was 2.4- to 3.2-fold higher than standard V̇_O2. Standard specific V̇_O2 had a body mass scaling exponent (b) of –0.18, and post-exercise specific V̇_O2 had a scaling exponent of –0.21 (Fig. 2). The fold increase in V̇_O2 due to exercise was also body mass specific, being greater in smaller animals. However, ANCOVA indicated that there was not a significant difference in the slope of the body mass dependence of standard and post-tail flip V̇_O2(F=0.46, P=0.50).

Fig. 3 illustrates the hypertrophic growth exhibited by the white abdominal musculature of the three species of crustaceans leading to larger fiber sizes in adults. Fiber size was proportional to body mass within and across species. Fibers in small P. pugio ranged from 35.6 to 130.9 μm, in large P. pugio from 47.3 to 150.3 μm, in small Penaeus spp. from 49.8 to 169.1 μm,in large Penaeus spp. from 31.7 to 258.1 μm and finally in small P. argus from 81.0 to 347.1 μm and in large P. argus from 18.6 to 317.2 μm. A Student's t-test indicated that, within each species, mean fiber sizes of small and large size classes were significantly different (Fig. 3, inset).

The mass-specific CS activity was evaluated according to the allometric scaling relationship, as described above. CS activity was independent of body mass, with a b value of 0.09 that was not significantly different from zero(Fig. 4).

The tail-flipping stimulation procedure elicited a burst escape response in the three species and size classes, although the number of tail-flips to exhaustion was species specific (see below). Fig. 5 shows the changes in AP concentration during the initial phase of recovery. As expected, muscles from smaller species recovered with a significantly higher rate than in larger species (ANCOVA: F=32.86, P<0.0001). However, within each species, there were no significant differences in recovery rate between the small and large size classes (ANCOVA: P. pugio, F=0.33, P=0.57; Penaeus spp., F=0.27, P=0.61; P. argus, F=0.05, P=0.82). Fig. 6 shows the scaling relationships with body mass of AP recovery, whole-animal post-exercise O₂ consumption, and muscle CS activity, using means for all data to correspond with the AP recovery data. The body-mass scaling exponent for AP recovery (b=–0.12) was less than whole-animal post-tail flip O₂ consumption (b=–0.21), but greater than muscle oxidative capacity, indexed by CS activity (b=0.09)(Fig. 6).

The lactate concentration was determined immediately after exercise and after 15 min of recovery. The lactate concentration immediately after contraction was not significantly different from the concentration 15 min after contraction for either size class in any of the three species(Fig. 7). However, limited animal availability for lactate assays in the large P. argus size class prevented statistical analysis within this group. Nevertheless,post-contractile anaerobic metabolism does not appear to be an important component of recovery in these species, despite a wide variation in animal body mass and fiber sizes. The contractile production of lactate varied between species mostly owing to the varying number of tail-flips performed during the exercise protocol (Fig. 7). This variation occurred despite the fact that all species tail-flipped a number of times that was equal to half the number that induced exhaustion (see Materials and methods).

Fig. 8 shows model calculations of the ATP turnover rate as a function of the rate constants governing ATP demand (k₁) and ATP supply(k₂) for the case without diffusion(Eqn 8, which defines the denominator in the calculation of η) for a given cell size and O₂ concentration at the boundary. This figure shows that, for a constant value of k₁, the reaction rate from the model increases with increasing k₂ until it asymptotically approaches a limiting value at very large k₂. Thus, it is clear that the ATP demand (k₁) sets an upper limit on the rate of ATP turnover. In addition, these results show that more than one combination of k₁ and k₂ can be used to give the experimentally observed rate of ATP turnover. The smallest value of k₁ at asymptotically large k₂ that matches the experimentally observed ATP turnover rate was determined for each condition of O₂ concentration at the boundary and cell size, and it was found (data not shown) that the highest average ATP and O₂concentrations occurred at this value of k₁.

Fig. 9A shows the dependence of the ATP turnover rate on the values of k₁ and k₂ for the case with diffusion(Eqn 9, which defines the numerator in the calculation of η). As with the no-diffusion case, many combinations of k₁ and k₂ can yield the experimentally observed ATP turnover rate. The minimum value of k₁ that would allow the model to satisfy the observed ATP turnover rate was determined by trial and error, and these values were slightly larger than the minimum k₁ values determined for the no-diffusion cases mentioned above. The corresponding k₂ values for these minimum values of k₁ were determined using a root-finding method. As in the cases without diffusion, it was found in the cases with diffusion that the lowest possible values of k₁ lead to the highest average ATP and O₂ concentrations. These cases also lead to the largest effectiveness factors, and Fig. 9B shows the corresponding relationship of k₁and k₂ to the effectiveness factor.

Our model calculations represent a conservative approach to the analysis of diffusion limitation, whereby kinetic parameters were identified that led to the highest possible values of η, and therefore the lowest possible extent of diffusion limitation. The values of the rate constants are not known in the present model; however, these values reflect the capacities for mitochondrial and ATPase reactions and are grounded in experimentally measured rates of AP recovery. Using this approach, intracellular concentration gradients are apparent in O₂, but not ATP, as is illustrated for the three O₂ concentrations for the smallest and largest fiber sizes in Fig. 10. It is implicit in the model utilized here that the ATP will have no or minimal concentration gradient because of the two no-flux boundary conditions for ATP and ADP. Despite the presence of oxygen concentration gradients, which have typically been used as a proxy for diffusion limitation, the influence of these gradients on ATP turnover is small at the high and intermediate O₂concentrations, as indicated by the relatively high effectiveness factors for all fiber sizes and experimentally observed AP recovery rates(Table 1). This means that the measured rate of AP recovery is too slow to be substantially limited by diffusion of oxygen or HEP metabolites. By contrast, effectiveness factors were consistently lower at the lowest O₂ concentration(Table 1). It should be noted that, for each species, diffusion limitation increased as the fibers grow larger. These developmental trajectories are presented graphically in Fig. 11, but, in this case, a single value of k₁ was used in order to capture the behavior of η for all species and size classes. Clearly, all species become more diffusion limited as they grow, and it is also noteworthy that here the effectiveness factors are much lower when k₁ and k₂ are not selected to maximize η, indicating substantial diffusion limitation in the adult animals.

The major finding of the present study was that whole-animal post-exercise V̇_O2 had a more negative body-mass scaling exponent (b) than post-contractile muscle AP recovery, and AP recovery, in turn, had a more negative scaling exponent than muscle CS activity. Post-contractile lactate accumulation was not significant,and anaerobic metabolism therefore does not appear to play a major role in the recovery rate of AP in the three groups examined. A mathematical reaction–diffusion model yielded relatively high effectiveness factors for all species except at the lowest hemolymph P_O2, indicating that the observed rates of muscle AP recovery might not be substantially limited by diffusion under most conditions. However, the decrease in effectiveness factors associated with increased fiber size suggests that at least some of the difference in the scaling of muscle aerobic capacity (indicated by CS activity) and an aerobic process (AP resynthesis) might be due to diffusion limitation.

In all species, fibers grew hypertrophically, and fiber size within and across species was proportional to body mass. Hypertrophic growth leads to developmental increases in diffusion distances that limit the permissible rates of intracellular aerobic processes(Kinsey et al., 2007). Changes in fiber dimensions occur despite the fact that the burst escape-and-recovery function of the muscles remained essentially the same in juvenile and adult animals. Owing to the greater diffusion distances, we expected to find increased diffusion limitation of aerobic processes in larger fibers. For instance, the time, t, required for a particle to traverse a given distance by diffusion is described by t=λ²/2D, where λ is the root-mean-square displacement and D is the diffusion coefficient. Therefore, the time required for O₂ or ATP to diffuse from the sarcolemma to the center of a fiber from the large size class of P. argus is approximately nine times longer than in a fiber from the small size class of P. pugio. Previous work on hypertrophically growing crustaceans and fish anaerobic fibers has shown that diffusion limitation of AP recovery is indeed highest in adult animals(Boyle et al., 2003; Kinsey et al., 2005; Nyack et al., 2007). These prior studies examined only HEP diffusion, and the authors suggested that low SA:V and associated limitations on the diffusion of O₂ across the sarcolemma might be more important than HEP diffusion in constraining the rate of aerobic recovery after burst-contractions (see below).

The standard metabolic rates measured in our current study were consistent with previous studies of species of Panulirus(Winget, 1969; Buesa, 1979; Diaz-Iglesias et al., 2004), Penaeus (Bishop et al.,1980; Dall, 1986; Villareal and Ocampo, 1993)and Palaeomonetes (MacFarland and Pickens, 1965) (Fig. 2). The observed increase in O₂ consumption after several bouts of exercise was expected. In both size classes of P. argus and Penaeus spp., the exercise regimen caused an approximately 2.3-fold increase in V̇_O2, whereas there was a 3.2-fold increase in V̇_O2 for the much smaller P. pugio. This is consistent with prior studies that have shown that increases in activity commonly cause a three- to five-fold increase in O₂ uptake in aquatic crustaceans(Taylor, 1982; Full and Herreid, 1983; Hamilton and Houlihan, 1992; McGraw, 2007), although a 10-fold increase in O₂ uptake after exercise has been found in the tail-flipping crayfish Pacifastacus lenisculus(Taylor, 1982). It should be noted, however, that the relatively small fold-increase in metabolic rate in the present study results in part from the fact that the increases in post-exercise V̇_O2were relative to standard V̇_O2, whereas some of the other cited studies compared the increase with basal V̇_O2. The scaling with body mass of post-exercise V̇_O2(b=–0.21) and standard V̇_O2(b=–0.18) was similar to that previously observed in crustaceans (MacFarland and Pickens,1965; Bridges and Brand,1980; Zoutendyk,1989; Martinez Palacios et al., 1996) (Fig. 2).

The rates of AP recovery were also similar to values previously reported for anaerobic muscle from fishes (Curtin et al., 1997; Hardewig et al.,1998; Nyack et al.,2007), mollusks (Bailey et al.,2003) and other crustaceans(Onnen and Zebe, 1983; Hansen et al., 1986; Morris and Adamczewska, 2002; Boyle et al., 2003). Data from the three species used in the present study had a combined body mass scaling exponent of –0.12 (Fig. 6), which is similar to the overall scaling exponent of–0.14 obtained when data from the present study were combined with published results for white muscle of similar aerobic capacity(Fig. 12). Likewise, the positive scaling with body mass of CS activity (b=0.09) is consistent with prior studies of anaerobic muscle. CS is a key regulatory enzyme in the citric acid cycle and gives a quantitative estimate of aerobic capacity(Berges and Ballantyne, 1991),although it should be noted that measures of activity might not necessarily reflect rates of in vivo flux through the reaction. CS activity in skeletal muscle often has an inverse relationship with body mass, but the slope is usually less steep than the slope for mass-specific standard metabolic rate (Childress and Somero,1990; Somero and Childress,1990; Emmett and Hochachka,1981). Other studies have found white muscle CS activity to scale positively with body mass, as in the present study(Berges and Ballantyne, 1991),or to have a slope near zero (Wells et al., 2001; Boyle et al.,2003; Nyack et al.,2007).

If diffusion is limiting metabolism in muscle, we would expect the scaling exponent for AP recovery to be more negative than that for V̇_O2 or CS activity as fiber size (and therefore diffusion distance) was proportional to body mass. Using this criterion, diffusion might account for the difference in scaling of AP recovery rate and CS activity, but not AP recovery rate and V̇_O2, as V̇_O2 had the most negative scaling exponent. In addition, if diffusion were limiting, we would expect to find evidence that the rate of an observed aerobic process (i.e. AP recovery) is, in fact, less than it would be if it were dependent on catalytic capacity alone. To address this issue, we developed a reaction–diffusion mathematical model to assess the extent to which diffusion might limit AP recovery in muscle and therefore possibly influence scaling behavior. The effectiveness factor (η) was employed here because it represents a means of evaluating the influence of diffusion on metabolic flux. We used a very conservative approach, where the kinetic properties that dictate ATP turnover, k₁ and k₂, were generated in a way that would yield the highest possible effectiveness factor. This resulted in high η values for the high and intermediate O₂ concentrations,whereas effectiveness factors were lower at the lowest O₂concentration. This suggests that, when animals are faced with environmental hypoxia or undergo intense exercise, a potential decrease in blood P_O2 might cause a greater diffusion limitation that possibly constrains the rate of metabolic recovery(Table 1; Fig. 11). This is consistent with the observation that reduced arterial P_O2during hypoxia can inhibit aerobic metabolic processes in crustacean muscle(Mende et al., 2003) despite a variety of compensatory responses designed to preserve O₂ delivery, such as an increased rate of ventilation and blood shunting to the muscle (McMahon,2001). In fact, the scaling exponent of η at the lowest P_O2 (b=–0.10) was equal to one half of the difference in b values between CS activity and AP recovery rate,lending some support to the notion that diffusion is partly responsible for the difference.

It is also clear from Fig. 10 that concentration gradients for O₂ are substantial in all cases, and gradients increase with either increasing fiber size or decreasing blood P_O2. This is consistent with prior suggestions that O₂ delivery is a greater constraint on muscle design and function than is HEP diffusion(Kinsey et al., 2005; Hardy et al., 2006; Nyack et al., 2007). This is in agreement with the notion that the shift in mitochondrial distribution towards the periphery of fibers that occurs during muscle growth in some crustacean and fish white muscle is a response to strong O₂gradients in the large fibers of adult animals(Boyle et al., 2003; Nyack et al., 2007). As we do not have data on mitochondrial distribution for the species used in the present study, we assumed a uniform distribution, which is consistent with previous measurements in blue crab muscle fibers of a size comparable to those analyzed here (Boyle et al.,2003). However, Mainwood and Rakusan(Mainwood and Rakusan, 1982)showed mathematically that a peripheral mitochondrial distribution reduced the intracellular concentration gradient for O₂ and led to better spatial buffering of ATP concentration across the fiber. Thus, a non-uniform organization of mitochondria might reduce diffusion effects. We also did not include in our analysis the possibility that arginine kinase is localized to different regions within the cell, and that this might reduce the magnitude of concentration gradients in HEP compounds. While data for arginine kinase localization in crustacean anaerobic fibers are limited, the role of phosphagen kinase localization on diffusion-dependent processes is typically considered to be important only in highly aerobic fibers (reviewed in Ellington, 2001). Furthermore,we found very slight gradients for HEP compounds when we assumed a homogeneous arginine kinase reaction, so the potential impact of increasing the model complexity to account for localization is negligible.

While some of the considerations described above imply that diffusion effects might be less than we reported, it is also plausible that diffusion limitation in muscle is more extreme than indicated in our analysis. For instance, the AP recovery rate might substantially underestimate total ATP turnover during recovery, which could lead to a dramatically increased effectiveness factor. Tail-flipping caused large metabolic perturbations leading to a 2.3- to 3.2-fold increase in whole-animal V̇_O2 above standard V̇_O2 due to activating the abdominal muscle group. However, we estimated the relative cost of AP recovery and found that the percentage of post-exercise V̇_O2 devoted to AP recovery was body size dependent and ranged from only 1% in small P. pugio to 17% in large P. argus(Table 2). In other words, only a small fraction of metabolic recovery following tail flipping is devoted to AP recovery, which suggests that other processes in the muscle, such as pH and ionic gradient restoration, lactate processing and glycogen resynthesis might be additional important sources of ATP demand(Full and Herreid, 1983).

A possible explanation for the observed differences in whole-animal V̇_O2 and AP recovery is that anaerobic metabolism might contribute more to AP recovery in large fibers than in small fibres, thus leading to a less-negative scaling exponent. We hypothesized that anaerobic metabolism would be used to speed up the recovery process in large animals to compensate for diffusion constraints associated with large fiber size. In the blue crab, for example, anaerobic metabolism is used to speed up AP recovery following exercise, as shown by the observation that large muscle fibers continue to accumulate large amounts of lactate after contraction, whereas lactate accumulation is minimal in small fibers (Johnson et al., 2004). This led to a higher than expected rate of AP recovery in the fibers of the largest blue crabs (Kinsey et al.,2005). However, there was no evidence of post-contractile lactate accumulation in the present study, suggesting that anaerobic metabolism is not invoked to accelerate recovery. The greater lactate production observed during contraction (at time zero after contraction; Fig. 7) is consistent with previous work on tail-flipping yabbies, which demonstrated positive allometry for contractile lactate production as well as anaerobic capacity(Baldwin et al., 1999).

In summary, we found different scaling exponents for whole-animal V̇_O2, muscle CS activity and muscle AP recovery in three groups of tail-flipping crustaceans. Muscle fibers grew hypertrophically and varied in mean diameter from 70 to 210μm, leading to substantially greater intracellular diffusion distances in larger animals. A mathematical model revealed that fibers became more diffusion limited as they got larger, particularly when hemolymph P_O2 was low. Diffusion might, therefore,account for some of the differences in scaling exponents, although in most cases diffusion limitation was minimal. The large differences in scaling exponents probably also reflect different mechanisms of control affecting each measured variable, which is consistent with the notion that scaling exponents of complex processes represent aggregate functions that reflect the scaling behavior of the underlying components(Darveau et al., 2002).

We thank Dr Barry Ache and Dr Juan Aggio for donating small P. argus; Captain Gerry Compeau, Dave Wells, Jay Souza, Steve Hall and Jennifer Dorton for collecting large P. argus; Clark Gray and Rodney Hayward for collecting penaeid shrimp; Dr M. Bailey from the University of Aberdeen, UK, for some of the data provided in Fig. 12; Dr Chris Finelli for helping with graphs and figures; and Mark Gay and Dr Dick Dillaman for their assistance with microscopy. This study was funded by National Science Foundation grants to S.T.K. (IOS-0316909 and IOS-0719123) and B.R.L.(IOS-0315883 and IOS-0718499), a National Institutes of Health grant to S.T.K.(NIAMS R15-AR052708) and a Sigma-Xi grant to A.G.J.