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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
The influence of oxygen and high-energy phosphate diffusion on metabolic scaling in three species of tail-flipping crustaceans
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
* Author for correspondence (e-mail: agj6818{at}uncw.edu)
Accepted 21 July 2008
 |
Summary |
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 TOP Summary INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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O2) and muscle
arginine phosphate (AP) resynthesis rate, as well as muscle citrate synthase
(CS) activity, in three groups of tail-flipping crustaceans. Two size classes
in each of three taxa (Palaemonetes pugio, Penaeus spp. and
Panulirus argus) were examined that together encompassed a
27,000-fold range in mean body mass. In all species, muscle fiber size
increased with body mass and ranged in diameter from 70±1.5 to
210±8.8 µm. Thus, intracellular diffusive path lengths for
O2 and HEP molecules were greater in larger animals. The body mass
scaling exponent, b, for post-tail flipping
O2
(b=–0.21) was not similar to that for the initial rate of AP
resynthesis (b=–0.12), which in turn was different from that of
CS activity (b=0.09). We developed a mathematical
reaction–diffusion model that allowed an examination of the influence of
O2 and HEP diffusion on the observed rate of aerobic flux in
muscle. These analyses revealed that diffusion limitation was minimal under
most conditions, suggesting that diffusion might act on the evolution of fiber
design but usually does not directly limit aerobic flux. However, both within
and between species, fibers were more diffusion limited as they grew larger,
particularly when hemolymph PO2 was low, which
might explain some of the divergence in the scaling exponents of muscle
aerobic capacity and muscle aerobic flux.
Key words: oxygen consumption, arginine phosphate, citrate synthase activity, aerobic metabolism, anaerobic metabolism, metabolic scaling, diffusion
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INTRODUCTION |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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). 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 following
contraction 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, Pi, 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 PO2
(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 O2
consumption, or
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 observed
post-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.
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MATERIALS AND METHODS |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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Whole-animal oxygen consumption
Closed-chamber respirometry was used to measure the standard and
post-contractile mass-specific rate of oxygen consumption
(O2).
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.
O2 was
continuously recorded using a Servogor 102 chart recorder until O2
in 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
O2 were half of
those previously measured as exhaustive work. Following the exercise protocol,
the chamber was again sealed, and post-contractile
O2 was measured.
Animals were allowed to consume 30% of the oxygen in the chamber before
measurements were stopped.
Muscle fiber diameter
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–1
MgSO4 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–1
MgCl2 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.
Muscle citrate synthase activity
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.
Muscle arginine phosphate recovery rate
During a burst exercise–recovery cycle, there is a reciprocal change
in AP and inorganic phosphate (Pi) 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
Pi, 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–1
PIPES, stored on ice for 10 min and centrifuged at 16,000 g
for 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 Pi.
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 L-lactate concentration
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).
Statistical analysis
The influence of body mass on specific
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.
Reaction–diffusion model
The mathematical model was developed for the system shown in
Fig. 1, where O2 is
supplied at a fixed blood stream concentration, C°, and it is transported
from the blood to the fiber through a barrier that represents the vascular
endothelium and cell membrane with a fixed resistance, 1/kmt.
O2 is consumed by a pseudo-homogeneous second-order reaction at the
mitochondria, with 6 moles of ADP forming 6 moles of ATP for every mole of
O2 by the overall reaction:
 | (1) |
 | (2) |
|
The ATPase is also assumed to be uniformly distributed through the domain
from x=0 to x=L. The one-dimensional molar species balances
for ADP, ATP and O2, valid in the region from x=0 to
x=L are given by:
 | (3) |
 | (4) |
The first boundary condition reflects the fact that ATP is not transported
out of the cell and there is symmetry about the cell center
(x=L). The second boundary condition describes the transport
of oxygen across the vascular walls and cell membrane by diffusion with a
linear driving force, where C° is the bloodstream concentration
of O2. The last boundary condition indicates that the oxygen
distribution is symmetric with respect to the center of the cell, as in the
case for ATP in the first boundary condition. The above system of equations is
solved using the boundary value differential equation solver `bvp5c' in MATLAB
version 7.5.0.342 (Mathworks, Lowell, MA, USA) to determine the spatially
dependent concentrations and to determine the flux at the boundary,
x=0, as well as the average concentrations of oxygen and ATP defined
by:
 | (5) |
The effectiveness factors are determined following methods discussed
previously (Locke and Kinsey,
2008). The effectiveness factor is defined as the ratio of the
rate of the reaction in the presence of diffusion to the rate of the reaction
in the absence of diffusion. It therefore can range from 0 (complete
limitation of reaction flux by diffusion) to 1 (no limitation by diffusion),
and it is a useful tool for assessing the extent of diffusion limitation. In
the absence of diffusion, Eqns 3
and 4 can be shown to give:
 | (6) |
 | (7) |
Eqn 6 leads to a quadratic
equation that can be solved easily for the non-dimensional ATP and oxygen
concentrations in the absence of diffusion, C1wo and
C2wo, respectively. All roots of the quadratic are real;
however, only one root is within the physical domain of the problem. The
reaction rates in the case without and with diffusion, respectively, are
determined from:
 | (8) |
 | (9) |
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 effectiveness factor (
), which again is the ratio of the rate of
reaction with diffusion to that in the absence of diffusion, is determined by
the ratio of Eqn 9 to
Eqn 8 and can be represented by:
 | (10) |
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 k2 that 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.
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RESULTS |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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Whole-animal oxygen consumption
The allometric scaling equation, y=aMb, 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 O2
was 2.4- to 3.2-fold higher than standard
O2. Standard
specific O2 had
a body mass scaling exponent (b) of –0.18, and post-exercise
specific O2 had
a scaling exponent of –0.21 (Fig.
2). The fold increase in
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
O2
(F=0.46, P=0.50).
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Muscle arginine phosphate recovery rate
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).
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Muscle L-lactate concentration
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).
|
Mathematical modeling
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 ) 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 O2
concentrations occurred at this value of k1.
|
). 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 k1
and k2 to the effectiveness factor.
|
, 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 O2
concentrations, 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 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 , indicating
substantial diffusion limitation in the adult animals.
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|
DISCUSSION |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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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
PO2, 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=
2/2D, where 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
O2, whereas
there was a 3.2-fold increase in
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 O2
were relative to standard
O2, whereas some
of the other cited studies compared the increase with basal
O2. The scaling
with body mass of post-exercise
O2
(b=–0.21) and standard
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).
|
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
O2, as
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 () 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 values for the high and intermediate O2 concentrations,
whereas effectiveness factors were lower at the lowest O2
concentration. This suggests that, when animals are faced with environmental
hypoxia or undergo intense exercise, a potential decrease in blood
PO2 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 PO2
during 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 at the lowest
PO2 (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 PO2. 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 O2
gradients 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
O2 above
standard 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
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
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
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 210
µm, 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
PO2 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).
|
Acknowledgments |
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References |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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