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First published online October 7, 2008
Journal of Experimental Biology 211, 3214-3225 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.020677

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

Ana Gabriela Jimenez^1,*, Bruce R. Locke² and Stephen T. Kinsey¹

¹ Department of Biology and Marine Biology, University of North Carolina Wilmington, 601 South College Road, Wilmington, NC 28403-5915, USA
² Department of Chemical and Biomedical Engineering, Florida State University, FAMU-FSU College of Engineering, Tallahassee, FL 32310-6046, USA

View larger version (5K): [in this window] [in a new window]   Fig. 1. Schematic of mathematical model showing one-dimensional spatial domain where the position in the cell, x, ranges from x=0 (sarcolemma) to x=L (fiber center). O2 is supplied at a constant concentration on the outside of a membrane that constitutes the vascular endothelium, extracellular space and sarcolemma (hatched area), and the myosin ATPase and mitochondrial reactions are distributed uniformly through the spatial domain.

View larger version (10K): [in this window] [in a new window]   Fig. 2. (A) Standard and (B) post-exercise O2 for all three species and size classes. The scaling exponent, b, was –0.18 for standard O2 and –0.21 for post-exercise O2 (r2=0.33 and 0.47, for standard O2 and post-exercise O2, respectively, P<0.05).

View larger version (22K): [in this window] [in a new window]   Fig. 3. Abdominal white muscle fiber diameter distribution and mean diameter (inset) from each size class of the three species of crustaceans, P. pugio, Penaeus spp. and P. argus. Fiber size was significantly different between size classes within each species (Student's t-test, P<0.05). Values are means ± s.e.m.

View larger version (6K): [in this window] [in a new window]   Fig. 4. Scaling with body mass of citrate synthase (CS) activity from abdominal white muscle of each size class and species of crustacean. The scaling exponent, b, was 0.09 (r2=0.11, P<0.05).

View larger version (14K): [in this window] [in a new window]   Fig. 5. The initial rate of post-tail flip arginine phosphate (AP) recovery for the small (broken lines) and large (unbroken lines) size classes. Linear regressions for small and large P. pugio resulted in r2 values of 0.36 and 0.43 and regression equations of y=0.92x+2.61 and y=0.81x+2.40, respectively. Small and large Penaeus spp. had r2 values of 0.60 and 0.23 and regression equations of y=0.78x+1.97 and y=0.68x+3.08, respectively. Small and large P. argus yielded r2 values of 0.68 and 0.42 and regression equations of y=0.34x–1.72 and y=0.38x+0, respectively. All regression slopes were significantly different from zero (P<0.05), but recovery rates between the small and large size classes were not significantly different within any species (see text).

View larger version (6K): [in this window] [in a new window]   Fig. 6. Comparison of the relationship between the scaling with body mass of whole-animal metabolic rate (O2), muscle aerobic capacity (CS activity) and a muscle metabolic process (AP resynthesis) following tail-flipping responses in three species of crustaceans. Data are reported as means ± s.e.m. All regression slopes were significantly different from zero except for CS activity (O2=32.0M–0.21, r2=0.95, P<0.05; AP resynthesis rate=1.22M–012, r2=0.77, P<0.05; CS activity=6.18M0.09, r2=0.68).

View larger version (18K): [in this window] [in a new window]   Fig. 7. Lactate levels immediately after exercise (black bars) and after 15 min of recovery (gray bars), as well as the relationship between contractile lactate production and number of tail flips. Lactate accumulation did not increase significantly after 15 min of recovery in any group. However, as only two specimens of the large P. argus were available for lactate measurement (one for each time point), a statistical analysis was not possible for this group (numbers in parentheses for large P. argus represent replicates within a single specimen). Values are means ± s.e.m.

View larger version (8K): [in this window] [in a new window]   Fig. 8. Mathematical model of reaction rate for the case without diffusion as a function of the two reaction rate constants with a fixed diffusion distance (L=104.5 µm) and O2 boundary concentration of 7.85 µM. The broken line is for the case of k1=0.000635 s–1, which satisfies the experimentally measured rate of AP recovery of 0.379 mmol l–1 min–1 in the limit of large k2.

View larger version (35K): [in this window] [in a new window]   Fig. 9. Example of mathematical modeling of reaction rate and effectiveness factor for the case with diffusion for various values of k1 and k2 with a fixed diffusion distance (L=104.5) and O2 boundary concentration of 7.85 µmol l–1. The experimentally measured rate of AP recovery of 0.379 mmol l–1 min–1 is indicated by the unbroken line, which shows the range of values of k1 and k2 that satisfy this rate (left panel), and the corresponding effectiveness factor (right panel). Values of k1 and k2 that satisfied the observed rate and maximized the effectiveness factor were then determined using a root-finding method (see text).

View larger version (12K): [in this window] [in a new window]   Fig. 10. Examples of reaction–diffusion model output for the smallest and largest fibers. (Top panel) O2 and ATP concentration profiles for muscle from small P. pugio, where the diffusion distance is 35.1 µm and the AP recovery rate is 0.811 mmol l–1 min–1. (Bottom panel) O2 and ATP concentration profiles for muscle from large P. argus, where the diffusion distance is 104.5 µm and the AP recovery rate is 0.379 mmol l–1 min–1. Other data from the model for all species and size classes are in Table 1.

View larger version (55K): [in this window] [in a new window]   Fig. 11. Effectiveness factor as functions of rate and diffusion distance compared with the experimental data. The ATPase rate constant k1 is fixed for all species and size classes, where k1=0.008 s–1. Here, k2 was varied to generate the various rates, and the intermediate O2 concentration of 7.85 µmol l–1 was used for both cases. The unbroken lines connect the juvenile and adults of each species, where the circles are P. pugio, the triangles are Penaeus spp., and the squares are P. argus. Note that decreases for each species as the animals grow.

View larger version (6K): [in this window] [in a new window]   Fig. 12. Scaling with body mass of the AP/PCr recovery rate in white muscle from crustaceans (Crangon crangon, Gecarcoidea natalis, Callinectes sapidus), fishes (Oncorhynchus mykiss, Salmo gairdnei, Pachycara brachycephalum, Zoarces viviparus, Squalus acanthias, and three size classes of Centropristis striata), mollusks (Placopecten magellanicus, Aequipecten percularis, Adamussium colbecki) and the crustaceans from our current study. The regression line is described by the equation, recovery rate=1.56M–0.14 (r2=0.16, P<0.05). The recovery rates and body mass data for other species were obtained from the published literature (Onnen and Zebe, 1983; Hansen et al., 1986; Curtin et al., 1997; Hardewig et al., 1998; Morris et al., 2002; Bailey et al., 2003; Johnson et al., 2004; Nyack et al., 2007; Livingstone et al. 1981; Wang et al., 1994; Richards et al., 2003).

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