A reaction-diffusion analysis of energetics in large muscle fibers secondarily evolved for aerobic locomotor function

Hemolymph supply to locomotor muscles of the ghost crab Ocypode quadrata

Ghost crabs are the fastest and most aerobically fit of the land crabs. The exceptional locomotory capacity of these invertebrate athletes seemingly depends upon effective coupling between the cardiovascular system and skeletal muscles, but how these systems are integrated has not been well defined. In the present study, we investigated the relationship between aerobic muscle fibers within the skeletal muscles used to power running and the blood vessels supplying these muscles. We used histochemical staining techniques to identify aerobic versus glycolytic fibers and to characterize membrane invaginations within the aerobic fibers. We also determined how the diameters of these two fiber types scale as a function of body size, across two orders of magnitude. Vascular casts were made of the blood vessels perfusing these muscles, and special attention was given to small, capillary-like vessels supplying the fibers. Finally, we injected fluorescent microspheres into the hearts of living crabs and tracked their deposition into different muscle regions to quantify relative hemolymph flow to metabolic fiber types. Collectively, these analyses demonstrate that ghost crab muscles are endowed with an extensive arterial hemolymph supply. Moreover, the hemolymph flow to aerobic fibers is significantly greater than to glycolytic fibers within the same muscles. Aerobic fibers are increasingly subdivided by membrane invaginations as crabs increase in size, keeping the diffusive distances relatively constant. These findings support a functional coupling between a well-developed circulatory system and metabolically active muscle fibers in these invertebrates.

Sensitivity analysis of reaction-diffusion constraints in muscle energetics

Theoretical and experimental studies of aerobic metabolism on a wide range of skeletal muscle fibers have shown that while all fibers normally function within the reaction control regime, some fibers operate near the transition region where reaction control switches to diffusion control. Thus, the transition region between reaction and diffusion control may define the limits of muscle function, and analysis of factors that affect this transition is therefore needed. In order to assess the role of all important model parameters, a sensitivity analysis (SA) was performed to define the parameter space where muscle fibers transition from reaction to diffusion control. SA, performed on a previously developed reaction-diffusion model, shows that the maximum rate for the ATPase reaction (V(max,ATPase)), boundary oxygen concentration in the capillary supply (O ₂⁰), the mitochondrial volume fraction (ε(mito)), and the diffusion coefficient of oxygen (DO ₂) are the most sensitive parameters affecting this transition to diffusion control. It is demonstrated that fibers are not limited by diffusion for slow reactions (V(max,ATPase)  < 25 mM/min), high oxygen supply for the capillaries (O ₂⁰ ≥ 35 µM), and large amounts of mitochondria (ε(mito) ≥ 0.1). These conditions are applicable to muscle cells spanning a very broad range of animals. Within the diffusion-controlled region, the overall metabolic rate and ATP concentrations have much higher sensitivity to the diffusion coefficient of oxygen than to the diffusion coefficients of the other metabolites (ATP, ADP, P(i)).

Reaction-diffusion constraints in living tissue: effectiveness factors in skeletal muscle design

A mathematical model was developed to analyze the effects of intracellular diffusion of O(2) and high-energy phosphate metabolites on aerobic energy metabolism in skeletal muscle. We tested the hypotheses that in a range of muscle fibers from different species (1) aerobic metabolism was not diffusion limited and (2) that fibers had a combination of rate and fiber size that placed them at the brink of substantial diffusion limitation. A simplified chemical reaction rate law for mitochondrial oxidative phosphorylation was developed utilizing a published detailed model of isolated mitochondrial function. This rate law was then used as a boundary condition in a reaction-diffusion model that was further simplified using the volume averaging method and solved to determine the rates of oxidative phosphorylation as functions of the volume fraction of mitochondria, the size of the muscle cell, and the amount of oxygen delivered by the capillaries. The effectiveness factor, which is the ratio of reaction rate in the system with finite rates of diffusion to those in the absence of any diffusion limitations, defined the regions where intracellular diffusion of metabolites and O(2) may limit aerobic metabolism in both very small, highly oxidative fibers as well as in larger fibers with lower aerobic capacity. Comparison of model analysis with experimental data revealed that none of the fibers was strongly limited by diffusion, as expected. However, while some fibers were near substantial diffusion limitation, most were well within the domain of reaction control of aerobic metabolic rate. This may constitute a safety factor in muscle that provides a level of protection from diffusion constraints under conditions such as hypoxia.

Molecules in motion: influences of diffusion on metabolic structure and function in skeletal muscle

Metabolic processes are often represented as a group of metabolites that interact through enzymatic reactions, thus forming a network of linked biochemical pathways. Implicit in this view is that diffusion of metabolites to and from enzymes is very fast compared with reaction rates, and metabolic fluxes are therefore almost exclusively dictated by catalytic properties. However, diffusion may exert greater control over the rates of reactions through: (1) an increase in reaction rates; (2) an increase in diffusion distances; or (3) a decrease in the relevant diffusion coefficients. It is therefore not surprising that skeletal muscle fibers have long been the focus of reaction-diffusion analyses because they have high and variable rates of ATP turnover, long diffusion distances, and hindered metabolite diffusion due to an abundance of intracellular barriers. Examination of the diversity of skeletal muscle fiber designs found in animals provides insights into the role that diffusion plays in governing both rates of metabolic fluxes and cellular organization. Experimental measurements of metabolic fluxes, diffusion distances and diffusion coefficients, coupled with reaction-diffusion mathematical models in a range of muscle types has started to reveal some general principles guiding muscle structure and metabolic function. Foremost among these is that metabolic processes in muscles do, in fact, appear to be largely reaction controlled and are not greatly limited by diffusion. However, the influence of diffusion is apparent in patterns of fiber growth and metabolic organization that appear to result from selective pressure to maintain reaction control of metabolism in muscle.

Scaling with body mass of mitochondrial respiration from the white muscle of three phylogenetically, morphologically and behaviorally disparate teleost fishes

White muscle (WM) fibers in many fishes often increase in size from <50 μm in juveniles to >250 μm in adults. This leads to increases in intracellular diffusion distances that may impact the scaling with body mass of muscle metabolism. We have previously found similar negative scaling of aerobic capacity (mitochondrial volume density, V(mt)) and the rate of an aerobic process (post-contractile phosphocreatine recovery) in fish WM. In the present study, we examined the scaling with body mass of oxygen consumption rates of isolated mitochondria (VO(2mt)) from WM in three species from different families that vary in morphology and behavior: an active, pelagic species (bluefish, Pomatomus saltatrix), a relatively inactive demersal species (black sea bass, Centropristis striata), and a sedentary, benthic species (southern flounder, Paralichthys lethostigma). In contrast to our prior studies, the measurement of respiration in isolated mitochondria is not influenced by the diffusion of oxygen or metabolites. V(mt) was measured in WM and in high-density isolates used for VO(2mt) measurements. WM V(mt) was significantly higher in the bluefish than in the other two species and VO(2mt) was independent of body mass when expressed per milligram protein or per milliliter mitochondria. The size-independence of VO(2mt) indicates that differences in WM aerobic function result from variation in V(mt) and not to changes in VO(2mt). This is consistent with our prior work that indicated that while diffusion constraints influence mitochondrial distribution, the negative scaling of aerobic processes like post-contractile PCr recovery can largely be attributed to the body size dependence of V(mt).

Energy metabolism and metabolic depression during exercise in Callinectes sapidus, the Atlantic blue crab: effects of the bacterial pathogen Vibrio campbellii

Callinectes sapidus (Rathbun), the Atlantic blue crab, commonly harbors low to moderate amounts of bacteria in hemolymph and other tissues. These bacteria are typically dominated by Vibrio spp., which are known to cause mortality in the blue crab. The dose-dependent lethality of an isolate of Vibrio campbellii was determined in crabs; the mean 48 h LD50 (half-maximal lethal dose) was 6.2×105 colony forming units g–1 crab. Injection of a sublethal dose of V. campbellii into the hemolymph of the crab resulted in a rapid and large depression (30–42%) of metabolic rate, which persisted for 24 h. Because gills are an organ of immune function as well as respiration, we were interested in how bacteria injected into the crab would affect the energetic costs associated with walking. Overall metabolism (aerobic and anaerobic) more than doubled in crabs walking for 30 min at 8 m min–1. The metabolic depression resulting from bacterial injection persisted throughout the exercise period and patterns of phosphagen and adenylate consumption within walking leg muscle were not affected by treatment. The ability of crabs to supply required energy for walking is largely unaffected by exposure to Vibrio; however, Vibrio-injected crabs are less aerobic while doing so. This depressed metabolic condition in response to bacteria,present during moderate activity, could be a passive result of mounting an immune response or may indicate an actively regulated metabolic depression. A compromised metabolism can affect the performance of daily activities, such as feeding and predator avoidance or affect the ability to cope with environmental stressors, such as hypoxia.

A skeletal muscle model of extreme hypertrophic growth reveals the influence of diffusion on cellular design

Muscle fibers that power swimming in the blue crab Callinectes sapidus are <80 microm in diameter in juveniles but grow hypertrophically, exceeding 600 microm in adults. Therefore, intracellular diffusion distances become progressively greater as the animals grow and, in adults, vastly exceed those in most cells. This developmental trajectory makes C. sapidus an excellent model for characterization of the influence of diffusion on fiber structure. The anaerobic light fibers, which power burst swimming, undergo a prominent shift in organelle distribution with growth. Mitochondria, which require O2 and rely on the transport of small, rapidly diffusing metabolites, are evenly distributed throughout the small fibers of juveniles, but in the large fibers of adults they are located almost exclusively at the fiber periphery where O2 concentrations are high. Nuclei, which do not require O2, but rely on the transport of large, slow-moving macromolecules, have the inverse pattern: they are distributed peripherally in small fibers but are evenly distributed across the large fibers, thereby reducing diffusion path lengths for large macromolecules. The aerobic dark fibers, which power endurance swimming, have evolved an intricate network of cytoplasmically isolated, highly perfused subdivisions that create the short diffusion distances needed to meet the high aerobic ATP turnover demands of sustained contraction. However, fiber innervation patterns are the same in the dark and light fibers. Thus the dark fibers appear to have disparate functional units for metabolism (fiber subdivision) and contraction (entire fiber). Reaction-diffusion mathematical models demonstrate that diffusion would greatly constrain the rate of metabolic processes without these developmental changes in fiber structure.

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

We examined the influence of intracellular diffusion of O(2) and high-energy phosphate (HEP) molecules on the scaling with body mass of the post-exercise whole-animal rate of O(2) consumption (V(O(2))) 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 microm. Thus, intracellular diffusive path lengths for O(2) and HEP molecules were greater in larger animals. The body mass scaling exponent, b, for post-tail flipping V(O(2)) (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 O(2) 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 P(O(2)) was low, which might explain some of the divergence in the scaling exponents of muscle aerobic capacity and muscle aerobic flux.

Diffusional constraints on energy metabolism in skeletal muscle

Aerobic metabolic flux depends on the diffusion of high-energy phosphate molecules (e.g., ATP and phosphocreatine) from the mitochondria to cellular ATPases, as well as the diffusion of other molecules (e.g., ADP, Pi) back to the mitochondria. Here, we develop an approach for evaluating the influence of intracellular metabolite diffusion on skeletal muscle aerobic metabolism through the application of the effectiveness factor (eta). This parameter provides an intuitive and informative means of quantifying the extent to which diffusion limits metabolic flux. We start with the classical approach assuming an infinite supply of substrate at the fiber boundary, and we expand this model to ultimately include nonlinear boundary and homogeneous reactions. Comparison of the model with experimental data from a wide range of skeletal muscle types reveals that most muscle fibers are not substantially limited by diffusion (eta close to unity), but many are on the brink of rather substantial diffusion limitation. This implies that intracellular metabolite diffusion does not dramatically limit aerobic metabolic flux in most fibers, but it likely plays a role in limiting the evolution of muscle fiber design and function.

The long and winding road: influences of intracellular metabolite diffusion on cellular organization and metabolism in skeletal muscle

A fundamental principle of physiology is that cells are small in order to minimize diffusion distances for O(2) and intracellular metabolites. In skeletal muscle, it has long been recognized that aerobic fibers that are used for steady state locomotion tend to be smaller than anaerobic fibers that are used for burst movements. This tendency reflects the interaction between diffusion distances and aerobic ATP turnover rates, since maximal intracellular diffusion distances are ultimately limited by fiber size. The effect of diffusion distance on O(2) flux in muscle has been the subject of quantitative analyses for a century, but the influence of ATP diffusion from mitochondria to cellular ATPases on aerobic metabolism has received much less attention. The application of reaction-diffusion mathematical models to experimental measurements of aerobic metabolic processes has revealed that the extreme diffusion distances between mitochondria found in some muscle fibers do not necessarily limit the rates of aerobic processes per se, as long as the metabolic process is sufficiently slow. However, skeletal muscle fibers from a variety of animals appear to have intracellular diffusion distances and/or fiber sizes that put them on the brink of diffusion limitation. Thus, intracellular metabolite diffusion likely influences the evolution of muscle design and places limits on muscle function.

Scaling of postcontractile phosphocreatine recovery in fish white muscle: effect of intracellular diffusion

In some fish, hypertrophic growth of white muscle leads to very large fibers. The associated low-fiber surface area-to-volume ratio (SA/V) and potentially long intracellular diffusion distances may influence the rate of aerobic processes. We examined the effect of intracellular metabolite diffusion on mass-specific scaling of aerobic capacity and an aerobic process, phosphocreatine (PCr) recovery, in isolated white muscle from black sea bass (Centropristis striata). Muscle fiber diameter increased during growth and was >250 mum in adult fish. Mitochondrial volume density and cytochrome-c oxidase activity had similar small scaling exponents with increasing body mass (-0.06 and -0.10, respectively). However, the mitochondria were more clustered at the sarcolemmal membrane in large fibers, which may offset the low SA/V, but leads to greater intracellular diffusion distances between mitochondrial clusters and ATPases. Despite large differences in intracellular diffusion distances, the postcontractile rate of PCr recovery was largely size independent, with a small scaling exponent for the maximal rate (-0.07) similar to that found for the indicators of aerobic capacity. Consistent with this finding, a mathematical reaction-diffusion analysis indicated that the resynthesis of PCr (and other metabolites) was too slow to be substantially limited by diffusion. These results suggest that the recovery rate in these fibers is primarily limited by low mitochondrial density. Additionally, the change in mitochondrial distribution with increasing fiber size suggests that low SA/V and limited O(2) flux are more influential design constraints in fish white muscle, and perhaps other fast-twitch vertebrate muscles, than is intracellular metabolite diffusive flux.