Multidimensional Conservation Laws: Overview, Problems, and Perspective

Some of recent important developments are overviewed, several longstanding open problems are discussed, and a perspective is presented for the mathematical theory of multidimensional conservation laws. Some basic features and phenomena of multidimensional hyperbolic conservation laws are revealed, and some samples of multidimensional systems/models and related important problems are presented and analyzed with emphasis on the prototypes that have been solved or may be expected to be solved rigorously at least for some cases. In particular, multidimensional steady supersonic problems and transonic problems, shock reflection-diffraction problems, and related effective nonlinear approaches are analyzed. A theory of divergence-measure vector fields and related analytical frameworks for the analysis of entropy solutions are discussed.Comment: 43 pages, 3 figure

Analysis of preconditioning and multigrid for Euler flows with low-subsonic regions

For subsonic flows and upwind-discretized, linearized 1-D Euler equations, the smoothing behavior of multigrid-accelerated point Gauss-Seidel relaxation is analyzed. Error decay by convection across domain boundaries is also discussed. A fix to poor convergence rates at low Mach numbers is sought in replacing the point relaxation applied to unconditioned Euler equations, by locally implicit “time”-stepping applied to preconditioned Euler equations. The locally implicit iteration step is optimized for good damping of high-frequency errors. Numerical inaccuracy at low Mach numbers is also addressed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41714/1/10444_2005_Article_BF02123476.pd

Diet-Independent Remodeling of Cellular Membranes Precedes Seasonally Changing Body Temperature in a Hibernator

Polyunsaturated fatty acids (PUFA) have a multitude of health effects. Their incorporation into membrane phospholipids (PL) is generally believed to depend directly on dietary influx. PL influence transmembrane protein activity and thus can compensate temperature effects; e.g. PL n-6 PUFA are thought to stabilize heart function at low body temperature (Tb), whereas long chain (>C18) n-3 PUFA may boost oxidative capacity. We found substantial remodeling of membranes in free-living alpine marmots which was largely independent of direct dietary supply. Organ PL n-6 PUFA and n-6 to n-3 ratios were highest at onset and end of hibernation after rapid increases during a brief transitional period prior to hibernation. In contrast, longer chain PL n-3 PUFA content was low at end of summer but maximal at end of hibernation. After termination of hibernation in spring, these changes in PL composition were rapidly reversed. Our results demonstrate selective trafficking of PUFA within the body, probably governed by a circannual endogenous rhythm, as hibernating marmots were in winter burrows isolated for seven months from food and external cues signaling the approaching spring. High concentrations of PL n-6 PUFA throughout hibernation are in line with their hypothesized function of boosting SERCA 2a activity at low Tb. Furthermore, we found increasing rate of rewarming from torpor during winter indicating increasing oxidative capacity that could be explained by the accumulation of long-chain PL n-3 PUFA. It may serve to minimize the time necessary for rewarming despite the increasing temperature range to be covered, because rewarming is a period of highest metabolic rate and hence production of reactive oxygen species. Considering the importance of PUFA for health our results may have important biomedical implications, as seasonal changes of Tb and associated remodeling of membranes are not restricted to hibernators but presumably common among endothermic organisms

Gluconeogenic adaptations in Cancer Magister

The periodic requirement for a new chitincus excskeleton imposes a large biosynthetic lead on the metabolism of crustaceans, with the hypodermis facing the brunt of the load. Since the freshly molted animal is highly susceptible to predation, the mechanisms for efficient gluccnecgenesis in support of chitin synthesis are of definite survival value to the organism. Measurements of enzyme activities in the hypodermis , gill and muscle of Cancer magisrter indicate that the hypodermis and muscle undergo considerable metabolic changes during the melt cycle. Freshmolt hypodermis shows elevated specific activities of both the gluconeogenic and the glycolytic enzymes, while freshmolt muscle shows decreased levels of the glycolytic and increased levels of the gluconeogenic enzymes. Hypodermis maintains a highly gluconeogenic orientation throughout the melt cycle. Phosphoglycerate kinase is considered to be one of the bifuncticnal enzymes in the glycolytic pathway, but the kinetic characteristics of the previously studied enzymes are ill suited for function in a gluconeogenic system. Since the inter melt and freshmolt muscle and hypodernis present a variety of metabolic poises (i. e. ranging 'from highly gluconeogenic tc highly glycolytic), I studied the control of phos|lidglycerate kinase in these tissue,s:. I found that the inte^0^^iusc 1 e enzyme shows kinetics much like those of the mammalian muscle and the yeast enzyme, with a high sensitivity to MgADP/ADP inhibition (MgADP Ki = 1.3 x 10⁻⁵M) and a relatively lew affinity for ATP as a substrate (Km = 2.03 x 10-⁴M). By contrast, the freshmolt hypodermal enzyme shows a considerably decreased sensitivity to Mg ADP/ADP inhibition (Mg ADP Ki = 2 x10-⁴M) and a considerably increased affinity for ATP (Km = 6.8 x 10-⁵M). The freshmolt muscle enzyme also shares these changed affinities. The intermolt hypodernal phosphoglycerate kinase shows the decreased sensitivity to Kg ADP/ADP inhibition but shares the ATP affinity of the intermolt muscle enzyme. The kinetic characteristics of the freshmolt hypodermal and muscle enzymes reduce the susceptibility of the enzymes to inhibition by MgADP and facilitate the reversal of the reaction for gluccneogenesis. The control of pyruvate kinase is integral to the control of both glycolysis and gluconeogenesis. In glycolysis, it forms the second major control site; in gluconeogenesis, it is one of the prime determinants of the rate of gluconeogenesis from lactate and amino acids. Muscle and hypodermal pyruvate kinases from Cancer magister are distinct proteins, on the basis of isoelectric points, kinetic characteristics, and thermal denaturation behavior. In contrast to the phcsphoglycerate kinase system, there are no pronounced differences between freshmolt and intermolt forms. Muscle pyruvate kinase is activated by FDP, inhibited by KgATP, arginine phosphate, Mg2citrate, tryptophan and is also sensitive to some inhibition by alanine, ⍺--glycerolphosphate, Mg-malate and ⍺- ketoglutarate. The muscle enzyme has a high affinity for PEP (Km = 0.1 mM) and the addition of 0.05 mM FDP drops the PEP Km to 0.05 mM. In comparison with other muscle pyruvate kinases, the enzyme is quite sensitive to MgATF inhibition (Ki = 1.8 mM) and shows FDP reversal of the inhibition. Arginine phosphate inhibition is competitive with ADP, and is not reversed by FDP. The reversal of the reaction accounts for only 0.5% of the forward reaction. Although high levels of ATP and arginine phosphate strongly inhibit the reaction, the inhibition is not sufficient to allow net flux through the low levels of the bypass enzymes present in the muscle of freshmolt animals. Thus, muscle pyruvate kinase has kinetic characteristics which suit it for function in the control of glycolysis, but do not allow gluconeogenic flux past the reaction locus. In contrast, hypodermal pyruvate kinase is a consplex protein capable of making large transitions between high activity during oxidation of carbohydrate substrates and virtually no activity during gluconeogenesis from lactate and amino acids. Hypodermal pyruvate kinase exists in twc conformational states, one a high affinity state (PK I) and the ether a low affinity state (PK II), PK I has a Km for PEP of 0.1 mM and a Ka for FDP of 1.3 x 10⁻⁵. PK II has a Km for PEP of 0.55 mM and a Ka for FDP of 9 x 10⁻⁸. For both forms, FDP facilitates the binding of PEP, Eofh forms are sensitive to MgATP inhibition and show FDP reversal of the inhibition, PK II is more sensitive to inhibition by alanine, serine, and Mg2citrate. For PK II, FDP alters the inhibition due to these compounds, changing the interactions between these inhibitors and both PEE and ADP. Incubation of PK II with 0.05 mK FDP produces PK I. Prolonged dialysis of PK I leads to an enzyme with the characteristics of PK II. The levels of FDP associated with PK I are higher than the levels associated with PK II. During gluconeogenesis, the FDP levels in the cell are low. This would shift the equilibrium between the two forms towards PK II. Since physiological levels of PEP, ADP, ATP, alanine, and serine limit PK II activity to less than 0.5% of maximal, considerable flux through the phosphcenolpyruvate carboxykinase and pyruvate carboxylase bypass would be feasible. The rise in FDP levels during inhibition of gluconeogenesis would shift the equilibrium in favor of PK I. This shift would immediately raise pyruvate kinase activity from less than 0.5% to around 50% of maximal activity. This, coupled with the other changes in metabolite levels during an inhibition of gluconeogenesis, would lead to a marked activation of pyruvate kinase activity. These conformational states allow rapid changes in flux through the reaction, and thus would allow flexible and responsive regulation of this important glycolytic and gluconeogenic control site. Thus, both the phcsphoglycerate kinase and pyruvate kinase present in the hypodermis of Cancer magister have special characteristics which facilitate efficient gluconeogenesis. To elucidate the possible importance of ions in regulating the activity of the above enzymes, I measured the levels of sodium, potassium, magnesium, and calcium in the muscle and hypodermis of intermolt and freshmolt animals. I found that the extracellular' space of- the hypodermis is considerably higher than that of the muscle (45% versus 12%) , but that there was no variation between intermolt and freshmolt tissue extracellular space. While freshmolt muscle sodium concentrations were significantly higher than intermolt sodium concentrations, none of the other ions showed significant differences between molt cycle stages. However, there were significant differences between the ionic concentrations in hypodermis and muscle. Hypodermis shewed higher calcium levels and lower potassium levels than muscle in both freshmolt and intermolt animals. Although icnic changes do not play a role in differential regulation of enzyme activity during the molt cycle, the ionic concentrations present in these tissues are such that the ions could set guidelines for the activity of phosphoglycerate kinase, pyruvate kinase, phosphofructokinase, and fructose diphosphatase in these tissues.Science, Faculty ofZoology, Department ofGraduat