Sugar transport in animal cells: the passive hexose transfer system

Cellular growth in biofilms

In this paper we develop a macroscopic evolutionary equation for the growth of the cellular phase starting from a microscopic description of mass transport and a simple structured model for product formation. The methods of continuum mechanics and volume averaging are used to develop the macroscopic representation by carefully considering the fluxes of chemical species that pertain to cell growth. The concept of structured models is extended to include the transport of reacting chemical species at the microscopic scale. The resulting macroscopic growth model is similar in form to previously published models for the transport of a single substrate and electron donor and for the production of cellular mass and exopolymer. The method of volume averaging indicates under what conditions the developed growth model is valid and provides an explicit connection between the relevant microscopic model parameters and their corresponding macroscopic counterparts.

A sugar transporter as a candidate for the outer hair cell motor

Forces developed by cochlear outer hair cells (OHCs) are responsible for the sharp tuning that underlies sensitivity and frequency selectivity in the ear. OHCs exhibit a voltage-dependent motility involving a 'motor' protein embedded in the basolateral membrane. The motor has so far resisted molecular identification. Here we provide evidence that it may be related to a fructose transporter. We show that OHCs are able to transport this sugar selectively and that the sugar alters electrical properties of the OHC motor. These data can be combined into an integrated model of a sugar carrier, that makes the novel prediction, demonstrated here, that such 'neutral' transporters can be voltage dependent.

Insulinomimetic effects of myricetin on lipogenesis and glucose transport in rat adipocytes but not glucose transport translocation

Myricetin is a naturally occurring flavonol that is commonly found in tea, berries, fruits, and medicinal plants. It mimics insulin in stimulating lipogenesis and glucose transport in rat adipocytes in vitro. It was found to stimulate lipogenesis in rat adipocytes and enhance the stimulatory effect of insulin. The EC50 was estimated to be about 65 microM. Myricetin did not have any effect on insulin receptor autophosphorylation nor on the tyrosine kinase activity of the receptor. However, myricetin stimulated both D-glucose and D-3-O-methylglucose uptake in rat adipocytes. The Vmax of glucose transport was increased, but the Km did not change significantly. Immunoblot analysis of Glut4 in rat adipocyte plasma membrane showed that the stimulation of glucose transport was not a consequence of glucose transporter translocation. Instead, the stimulation in glucose uptake probably was due to a change in the intrinsic activity of the glucose transporter possibly caused by alterations in membrane fluidity or transporter-lipid interactions as a result of the insertion of myricetin into the membrane bilayer. Thus, myricetin may have therapeutic potential in the management of non-insulin-dependent diabetes mellitus by stimulating glucose uptake without the presence of fully functional insulin receptor.

Glucose transport in amastigotes and promastigotes of Leishmania mexicana mexicana

Promastigotes and amastigotes of Leishmania mexicana mexicana transported 2-deoxy-D-glucose (2-DOG) by a saturable process with a Km of 24 +/- 3 microM and Vmax of 2.21 nmol min-1 (mg protein)-1 for the promastigote and a Km of 29 +/- 8 microM and Vmax of 0.13 nmol min-1 (mg protein)-1 for the amastigote stage. Amastigotes incorporated 2-DOG maximally at pH 5.0, while for promastigotes the optimum was at pH 7.0. Mid-log phase promastigotes were found to accumulate 2-DOG via a stereospecific carrier-mediated process which was competitively inhibited by D-glucose and D-mannose but not L-glucose. Transport was dependent upon temperature, with a Q10 in promastigotes of 1.83 and an optimum rate at 35 degrees C (+/- 4 degrees C) with an activation energy of 50.12 kJ mol-1. Stationary phase promastigotes accumulated 2-DOG at approximately twice the rate of mid-log phase promastigotes. Cytochalasin B, forskolin and phloretin were all found to inhibit human erythrocyte 2-DOG uptake but only cytochalasin B was found significantly to inhibit promastigote 2-DOG uptake. Interestingly, leishmanial 2-DOG uptake was inhibited by a series of membrane potential antagonists including the ionophore monensin, the H+ATPase inhibitor N, N'-dicyclohexylcarbodiimide (DCCD) and uncoupling agent carbonylcyanide-4-(triflouromethoxy) phenylhydrazone (FCCP), as well as, the tricyclic drugs chlomipramine and imipramine, but was insensitive to the Na+/K+ATPase inhibitor ouabain and the antitrypanosomal drugs Pentostam and Suramin. We therefore conclude that there are significant structural and mechanistic differences between the D-glucose uptake systems of Leishmania and the mammalian host to merit the inclusion of glucose transporters as putative targets for rational drug design.

Elevated cytosolic calcium levels in human lymphocytes during surface virus infections

Generalised metabolic and electrolyte disturbances are known to accompany both plasma and surface virus infections. We have investigated whether these infections could impair the transport of Ca2+ from cells under conditions of controlled concentrations of the energy substrate glucose. Thus, cytosolic calcium levels ([Ca2+]i) were measured in single isolated lymphocytes obtained from healthy volunteers or those suffering from coryza. Before making measurements using a Ca(2+)-sensitive fluorescent dye indo 1, we incubated lymphocytes in buffers containing 0 mM-, 5.6 mM- or 11.2 mM-[glucose]. We found that [Ca2+]i of lymphocytes obtained from the sick were significantly higher than those from healthy controls both at 0 mM and 5.6 mM-[glucose], and that [Ca2+]i was inversely related to the media glucose concentration for both groups. These results suggest a diminished capacity of cation pumping in viral infections, such as coryza, in relationship to the available glucose as energy substrate.

Glycosylation of the human erythrocyte glucose transporter: a minimum structure is required for glucose transport activity

The involvement of the carbohydrate moiety of the human erythrocyte glucose transporter in glucose transport activity was previously demonstrated (Feugeas et al. (1990) Biochim. Biophys. Acta 1030, 60-64): N-glycanase treatment of the transport glycoprotein reconstituted in proteoliposomes resulted in a dramatic decrease of the Vmax. In this study, kinetic measurements of glucose equilibrium influx confirm our previous results. In order to investigate that a minimum glycosidic structure is required to maintain glucose transport activity, proteoliposomes were respectively treated with either sialidase, or sialidase and endo-beta-galactosidase, or a pool of exo-glycosidases which allows the release of all the sugar residues, except the proximal N-acetylglucosamine. Kinetic measurements of zero-trans influx made on sialidase- and (sialidase + endo-beta-galactosidase)-treated proteoliposomes did not reveal any significant changes in the glucose transport activity. On the contrary, treatment of the same proteoliposomes by a pool of exoglycosidases led to a complete abolition of activity, suggesting that a minimum glycosidic structure is required for glucose transport activity.

Effects of insulin on glucose uptake in cultured cells from the central nervous system of rodent

1. Incubation of C6 glioma cultures with insulin resulted in a time and dose-dependent stimulation of 2-deoxy-D-glucose uptake. The maximal stimulation (160% of the control) was observed with 1 nM insulin and 0.05 nM caused half-maximum effect. 2. Incubation of NG 108-15 (neuroblastoma x glioma hybrid) and N2 neuroblastoma cells with 160 nM insulin did not result in a significant stimulation of this glucose uptake. 3. The basal level and stimulatory effect by insulin on this glucose uptake observed in C6 glioma cells were dependent on the presence of calcium in the medium. 4. Such an increase in glucose uptake in C6 glioma cells was also observed in the presence of diacylglycerol (DG) generating agents, such as carbachol (1 mM) and phospholipase C (0.05 unit/ml) or of DG analogs, such as sn-1,2-dioctanoyl glycerol (250 microM) and phorbol myristate acetate (1 microM). 5. Our results indicated that both calcium ion and DG levels play important roles in the regulation of glucose uptake in the glial cells, but not in neuronal cells from the brain.

Glycosylation of the human erythrocyte glucose transporter is essential for glucose transport activity

The human erythrocyte glucose transporter is a fully integrated membrane glycoprotein having only one N-linked carbohydrate chain on the extracellular part of the molecule. Several authors have suggested the involvement of the carbohydrate moiety in glucose transport, but not definitive results have been published to date. Using transport glycoproteins reconstituted in proteoliposomes, kinetic studies of zero-trans influx were performed before and after N-glycanase treatment of the proteoliposomes: this enzymatic treatment results in a 50% decrease of the Vmax. The orientation of transport glycoproteins in the lipid bilayer of liposomes was investigated and it appears that about half of the reconstituted transporter molecules are oriented properly. Finally, it could be concluded that the release of the carbohydrate moiety from the transport glycoproteins leads to the loss of their transport activity.

Inositol uptake in rat aorta

The purpose of this study was to investigate the mechanism of inositol uptake into rat thoracic aorta. 3H-inositol uptake into deendothelialized aorta was linear for at least 2 h and was composed of both a saturable, Na(+)-dependent, and a nonsaturable, Na(+)-independent component. The Na(+)-dependent component of inositol uptake had a Km of 50 microM and a Vmax of 289 pmol/mg prot/h. Exposure to LiCl, ouabain, or Ca2(+)-free Krebs-Ringer bicarbonate solution inhibited uptake. Metabolic poisoning with dinitrophenol, as well as incubation with phloretin, an inhibitor of carrier-mediated hexose transport, also inhibited uptake. Exposure to norepinephrine decreased inositol uptake, while phorbol myristate acetate was without effect. Isobutylmethylxanthine significantly increased inositol uptake, while the increased uptake due to dibutyryl cyclic AMP and forskolin were not statistically significant. Sodium nitroprusside, an activator of guanylate cyclase, and 8-bromo cyclic GMP, were without effect on uptake, as was methylene blue, an inhibitor of guanylate cyclase. Inositol uptake into the aorta was increased when the endothelium was allowed to remain intact, although this effect was likely due to uptake into both the endothelial and smooth muscle cells. These results suggest that the uptake of inositol into vascular smooth muscle is: (1) dependent upon an inward Na(+)-gradient; (2) carrier mediated, and (3) inhibited by alpha 1 adrenoceptor agonists.

In vitro glucose and 2-aminoisobutyric acid uptake by rat interscapular brown adipose tissue

The dependence upon substrate and insulin concentrations, as well as on sodium and potassium concentrations in the medium of the uptake of glucose and 2-aminoisobutyric acid, was determined for fragments of brown and white adipose tissues incubated in vitro. Brown adipose tissue showed a high capacity for glucose uptake at high glucose concentrations, this uptake being dependent on both glucose and insulin concentration. White adipose tissue showed much more limited uptake capabilities. The presence of Na+ and K+ had little effect on the uptake. The uptake of 2-aminoisobutyric acid was similar in both adipose tissues, being enhanced by physiological levels of insulin and depressed by ouabain. This amino acid transport was dependent on Na+ and K+ concentrations, and the overall transporting capability was two to three orders of magnitude lower than that for glucose. It was concluded that amino acids could not play a significant role as bulk thermogenic substrates for brown adipose tissue, as their transporters lack the plasticity of response to high substrate and insulin concentrations which characterize brown adipose tissue uptake of glucose.

Characterization of low- and high-affinity glucose transports in the yeast Kluyveromyces marxianus

Glucose transport in the yeast Kluyveromyces marxianus proceeds by two functionally and presumably structurally distinct transporters depending on the carbon source of the culture medium. In lactose-grown cells, glucose was taken up through a high-affinity H+-sugar symporter (Km = 0.09 mM), whereas a low-affinity transporter (Km = 3.5 mM) was utilized in glucose-grown cells. The two transporters exhibited different substrate specificities. Galactose was demonstrated to be a selective substrate of the H+-glucose symporter (Km = 0.14 mM) and did not significantly enter glucose-grown cells. Fructose was a preferential substrate of the low-affinity carrier (Km = 3.5 mM), but it entered lactose-grown cells through a high-affinity H+-fructose symporter distinct from the H+-glucose one. Other putative substrates of the two glucose transporters were identified by competition experiments. 2-Deoxyglucose recognized both carriers with a similar affinity, while the non-phosphorylatable analogues 6-deoxyglucose, 3-O-methylglucose and D-fucose exhibited a 10-30 fold preference for the high-affinity transporter.

A single half-turnover of the glucose carrier of the human erythrocyte

Single half-turnovers of the glucose carrier of the human erythrocyte have been measured by recruiting carriers to the outward-facing conformation by (a) pre-exposing cells to extracellular maltose, or (b) pre-warming cells to 38 degrees C, before addition of D-[14C]glucose at 0 degrees C. Based on these experiments estimates of the number of glucose carriers per red cell range from 124,000 to 190,000.

Characterization of glucose transport in an insulin-secreting cell line

The rat insulinoma-derived RINm5F cell line retains many differentiated functions of islet beta-cells. However, it fails to recognize glucose as an insulin secretagogue in the physiological concentration range. With this cell line, glucose-transport kinetics were investigated, by using a double-label technique with the non-metabolizable glucose analogue 3-O-methylglucose (OMG). RINm5F cells possess a passive glucose-transport system with high capacity and low affinity. Equilibration across the plasma membrane of extracellular OMG concentrations up to at least 20 mM is achieved within 2 min at 37 degrees C. The half-saturation of OMG uptake occurs at 32 mM. At lower temperatures OMG uptake is markedly retarded, with a temperature coefficient (Q10) of 2.9. As indicated by efflux measurements, transport is symmetrical. Cytochalasin B at micromolar concentrations and phlorrhizin in millimolar concentrations are potent inhibitors of OMG uptake. Neutralization of the secreted insulin with antibodies does not alter OMG uptake kinetics. The glucose metabolism of RINm5F cells is much exaggerated compared with that of islet beta-cells. Nonetheless, when measured in parallel to uptake, transport exceeds by far the rate of metabolism at glucose concentrations above 3 mM. Measurements of intracellular D-glucose reveal a lower intracellular glucose concentration relative to the extracellular in RINm5F cells. This seems to be due to abnormalities in the subsequent steps of glucose metabolism, rather than to abnormalities in hexose uptake. The loss of glucose-induced insulin release in RINm5F cells cannot be explained by alterations in hexose transport.

Cytosolic free calcium concentration and glucose transport in isolated cardiac myocytes

The role of cytosolic free Ca2+ concentration, [Cai], in mediating insulin's stimulatory effect on glucose transport was investigated in isolated Ca2+-tolerant rat ventricular cells. Approximately 98% of glucose transport in isolated myocytes was inhibited by phloretin. Insulin-accelerated glucose transport by 50-115% over basal transport rate. Removal of extracellular Ca2+ had no effect on either the basal transport rate or insulin's stimulatory action, indicating that extracellular Ca2+ was not necessary for insulin's effect to be manifest. Addition of A23187 had no effect on glucose transport rate. Under basal conditions, [Cai] was 167 +/- 12 nM as measured by fura-2 fluorescence and 239 +/- 22 nM by null-point titration with arsenazo III. Loading cells with fura-2 did not affect basal glucose transport rates. In addition, the stimulatory effect of insulin on glucose transport was preserved in fura-2 loaded cells. In paired experiments, insulin did not increase [Cai] as measured by fura-2 fluorescence or null-point titration despite acceleration of glucose transport. In contrast, addition of KCl (40 mM) increased [Cai] from 168 +/- 30 to 287 +/- 51 nM and resulted in 50% reduction in glucose transport rate. In other experiments designed to control for the hyperosmolar effects of KCl, NaCl (40 mM) caused no change in [Cai] but also inhibited glucose transport rate by 50%. We conclude that an elevation in [Cai] is unlikely to be the intracellular signal mediating insulin's effect on glucose transport since insulin's stimulatory effect was not reduced by Ca2+ -free media, insulin had no detectable effect on [Cai], and elevation of [Cai] by KCl did not result in stimulation of glucose transport.

Kinetics of glucose transport in human erythrocytes: zero-trans efflux and infinite-trans efflux at 0 degree C

The kinetic features of glucose transport in human erythrocytes have been the subject of many studies, but no model is consistent with both the kinetic observations and the characteristics of the purified transporter. In order to reevaluate some of the kinetic features, initial rate measurements were performed at 0 degree C. The following kinetic parameters were obtained for fresh blood: zero-trans efflux Km = 3.4 mM, Vmax = 5.5 mM/min; infinite-trans efflux Km = 8.7 mM, Vmax = 28 mM/min. For outdated blood, somewhat different parameters were obtained: zero-trans efflux Km = 2.7 mM, Vmax = 2.4 mM/min; infinite-trans efflux Km = 19 mM, Vmax = 23 mM/min. The Km values for fresh blood differ from the previously reported values of 16 mM and 3.4 mM for zero-trans and infinite-trans efflux, respectively (Baker, G.F. and Naftalin, R.J. (1979) Biochim. Biophys. Acta 550, 474-484). The use of 50 mM galactose rather than 100 mM glucose as the infinite-trans sugar produced no change in the infinite-trans efflux Km values but somewhat lower Vmax values. Simulations indicate that initial rates were closely approximated by the experimental conditions. The observed time courses of efflux are inconsistent with a model involving rate-limiting dissociation of glucose from hemoglobin (Naftalin, R.J., Smith, P.M. and Roselaar, S.E. (1985) Biochim. Biophys. Acta 820, 235-249). The results presented here support the adequacy of the carrier model to account for the kinetics.

Active transport of myo-inositol in rat pancreatic islets

myo-Inositol transport by isolated pancreatic islets was measured with a dual isotope technique. Uptake was saturable with a half-maximal response at approx. 75 microM. With 50 microM-inositol, uptake was linear for at least 2 h during which time the free intracellular concentration rose to double that of the incubation medium. Inositol transport is therefore active and probably energized by electrogenic co-transport of Na+ down its concentration gradient as uptake was inhibited by ouabain, Na+ removal or depolarizing K+ concentrations. Inositol transport was abolished by cytochalasin B which binds to hexose carriers, but not by carbamoylcholine or Li+ which respectively stimulate or inhibit phosphoinositide turnover. Uptake of inositol was not affected by 3-O-methylglucose or L-glucose (both 100 mM) nor by physiological concentrations of D-glucose. The results suggest that most intracellular inositol in pancreatic islets would be derived from the extracellular medium. Since the transport mechanism is distinct from that of glucose, inositol uptake would not be inhibited during periods of hyperglycaemia.

Galactose uptake by human platelets in vitro

Galactose transport by human platelets has been studied by measuring the cellular accumulation of the radiolabeled sugar during brief periods of suspension in varying concentrations of galactose. Weighted least-squares regression curves fitted to the measurements (initial velocity versus galactose concentration) indicate that a kinetic model with two saturable components is statistically more consistent with the data than a model based upon a single process (P less than 0.001). For the two-component model Km1 = 0.29 mM, V1 = 1.2 mmol/min per 10(15) platelets, Km2 = 46 mM, V2 = 117 mmol/min per 10(15) platelets. The fact that galactose metabolites did not accumulate during the initial phase of uptake indicates that the uptake process is not mediated by enzymatic catalysis. Surface binding also appears inadequate to explain the uptake. The most likely basis for the kinetic data, therefore, is membrane transport. The kinetics are consistent with transport by coexistent membrane structures as well as with transport by a single structure manifesting negative cooperativity.

Experimental analysis of ion/solute cotransport by substrate binding and facilitated diffusion

An ion/solute cotransporter can be studied in the absence of a transmembrane gradient of the electrochemical potential of the ion. Inspection of the appropriate equations discloses that basic parameters of the cotransport cycle can be obtained by measuring cosubstrate binding and the initial-velocity kinetics of four modes of facilitated diffusion as a function of the concentration of the cotransported ion. The following information can be derived: estimates of the affinities of both cosubstrates, the number of binary intermediates participating in cotransport (equivalent to determining the order of cosubstrate binding and release), and the rate constants for the reorientation of the binding sites during cotransport. In general, both maximal velocities and half-saturation constants for the facilitated diffusion of one cosubstrate depend upon the concentration of the other. In some cases, the maximal velocities of influx and efflux do not increase monotonically with the concentration of the ion but pass through a maximum and decrease. If direct binding studies are not possible, affinities of the cosubstrates can be estimated from data for equilibrium exchange or countertransport. Also, an approximate description of the time course of the transient accumulation (overshoot) during countertransport is derived. Under certain circumstances, the height of the overshoot is proportional to the concentration of the cotransported ion. Thus, countertransport can be employed as a simple test to establish if a solute is cotransported with a particular ion. This treatment allows many effects noted in galactoside countertransport in Escherichia coli to be explained.

Facilitated glucose transporter of human erythrocyte: proteolytic mapping of the [3H]cytochalasin B photoaffinity-labeled transporter polypeptide

The human erythrocyte D-glucose transporter is an integral membrane glycoprotein with an heterogeneous molecular mass spanning a range 45-70 kDa. The protein structure of the transporter was investigated by photoaffinity labeling with [3H]cytochalasin B and fractionating the labeled transporter according to molecular mass by preparative SDS-polyacrylamide gel electrophoresis. Each fraction was digested with either papain or S. aureus V8 proteinase, and the labeled proteolytically derived peptide fragments were compared by SDS polyacrylamide gel electrophoresis. Papain digestion yielded two major peptide fragments, of approx. molecular mass 39 +/- 2 and 22 +/- 2 kDa; treatment with V8 proteinase resulted in two fragments, with mass of 24 +/- 2 and 15 +/- 2. Proteolysis of each transporter fraction produced the same pattern of labeled peptide fragments, irrespective of the molecular mass of the original fractions. The binding characteristics of [3H]cytochalasin-B-labeled transporter to Ricinis communis agglutinin lectin was examined for each transporter molecular mass fraction. It was found that higher-molecular-mass fractions of intact transporter had a 2-fold greater affinity for the lectin than lower-molecular-mass fractions (i.e., 67 kDa greater than 45 kDa fraction). However, proteolytically derived labeled peptide fragments from each fraction had minimal affinity for the lectin. These results suggest that the labeled peptide fragments have been separated from the glycosylated regions of the parent transporter protein. The present findings indicate that, although transporter proteins have an apparently heterogeneous molecular mass, some regions of the protein share a common peptide. Furthermore, the glycosylated regions appear to be located some distance from the [3H]cytochalasin-B-labeled site(s).

Distinction of three types of D-glucose transport systems in animal cells

Immunoblotting of plasma membrane fractions from rat kidney cortex with antibody to human erythrocyte glucose transporter showed a single major cross-reacting material of 48K in basolateral membrane fractions possessing a facilitated diffusion system for D-glucose, but not in brush border membrane fractions which have a Na-dependent active transport system. Cytochalasin B inhibited D-glucose uptake in basolateral membrane vesicles but not in brush border vesicles. Cross-reacting materials of 44-55K were detected in several animal cells exhibiting facilitated diffusion systems, including a hormone dependent system. These results indicate molecular difference between glucose transporters of facilitated diffusion systems and active transport systems.

Reconstitution of the glucose transporter from bovine heart

Reconstitution of the glucose transporter from heart should be useful as an assay in its purification and in the study of its regulation. We have prepared plasma membranes from bovine heart which display D-glucose reversible binding of cytochalasin B (33 pmol sites/mg protein; Kd = 0.2 muM). The membrane proteins were reconstituted into liposomes by the freeze-thaw procedure. Reconstituted liposomes showed D-glucose transport activity which was stereospecific, saturable and inhibited by cytochalasin B, phloretin, and mercuric chloride. Compared to membrane proteins reconstituted directly, proteins obtained by dispersal of the membranes with low concentrations of cholate or by cholate solubilization showed 1.2- or 2.3-fold higher specific activities for reconstituted transport, respectively. SDS-polyacrylamide gel electrophoresis followed by electrophoretic protein transfer and labeling with antisera prepared against the human erythrocyte transporter identified a single band of about 45 kDa in membranes from both dog and bovine hearts, a size similar to that reported for a number of other glucose transporters in various animals and tissues.

Asymmetric transport of a fluorescent glucose analogue by human erythrocytes

A fluorescent glucose analogue, 6-deoxy-N-(7-nitrobenz-2-oxa-1,3-diazol-4-yl)-aminoglucose (NBDG), was synthesized and its interactions with the hexose transport system of the human red blood cell were investigated. NBDG entry is inhibited by increasing concentrations of D-glucose (Ki = 2 mM). However, NBDG exit is unaffected by D-glucose in red blood cells. Cytochalasin B was found to inhibit both NBDG entry and exit. NBDG accumulates in the red blood cell above the theoretical equilibrium concentration. Accumulation of NBDG is temperature-sensitive and is due to the binding of NBDG to some intracellular substance. The binding of NBDG to purified hemoglobin suggests that accumulation of NBDG by erythrocytes is due to the intracellular binding of NBDG to hemoglobin. NBDG does not accumulate in pink erythrocyte ghosts, while its rate of uptake is still inhibited by D-glucose and cytochalasin B. Although there was no apparent D-glucose inhibition of NBDG exit by intact red blood cells, D-glucose was able to inhibit NBDG exit by pink erythrocyte ghosts. The differing properties of NBDG influx and efflux support the interpretation that the hexose transport system of the human red blood cell appears asymmetric although it may be intrinsically symmetric.