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We present two sets of qualgebras involving operators which generalize creation and annihilation operators. These two groups of operators satisfy separately quommutation relations rather than commutation or anticommutation relations. The quommutators of the creation and annihilation operators generate new “neutral operators” which themselves are subjected to quommutation relations. Two solutions are presented. In the second one, some new symmetry relations are added to the system. In a certain sense these extra relations, rather than imposing new constraints on the parameters, increase their freedom.

In rat liver homogenates fortified with the appropriate cofactors (ATP and CoA), valproic acid induced H2O2production rates by far lower than those recorded on the straight medium‐chain fatty acid n‐octanoic acid. Using directly the CoA esters of these carboxylic acids as substrates for the rat liver H2O2‐generating enzyme activities, valproyl‐CoA, and n‐octanoyl‐CoA were found to induce similar oxidation rates. In the rat liver homogenates, cyanide‐insensitive valproyl‐CoA and octanoyl‐CoA oxidations occurred at rates similar to those of valproyl‐CoA and octanoyl‐CoA oxidase(s), respectively. Studies on fractions obtained from rat liver postnuclear supernatants by isopycnic centrifugation on a linear sucrose density gradient disclose that the density distribution of valproyl‐CoA oxidase superimposes to those of catalase, fatty acyl‐CoA oxidase and cyanide‐insensitive fatty acyl‐CoA oxidation, three peroxisomal marker activities. By contrast, the cyanide‐insensitive valproyl‐CoA oxidation does not adopt the typical peroxisomal distribution of these activities but rather exhibits a mitochondrial localization with, however, a minor peroxisomal component. Interestingly enough, the comparative study of rat tissue distribution, inducibility by clofibrate and sensitivity to deoxycholate indicated that valproyl‐CoA oxidase is an enzyme distinct from fatty acyl‐CoA oxidase and bile acyl‐CoA oxidase. Taken as a whole, the results presented here support the occurrence of a peroxisomal oxidation of the CoA ester of valproic acid and its Δ4‐enoic derivate which might be characterized by two major features: initiation by an acyl‐CoA oxidase distinct from fatty and bile acyl‐CoA oxidases, and inability to complete the β‐oxidation cycle which would not proceed, at significant rates, further than the β‐hydroxyacyl‐CoA dehydrogenation step in peroxisomes.

In rat liver homogenates fortified with the appropriate cofactors (ATP and CoA), valproic acid induced H 2O 2production rates by far lower than those recorded on the straight medium-chain fatty acid n-octanoic acid. Using directly the CoA esters of these carboxylic acids as substrates for the rat liver H 2O 2-generating enzyme activities, valproyl-CoA, and n-octanoyl-CoA were found to induce similar oxidation rates. In the rat liver homogenates, cyanide-insensitive valproyl-CoA and octanoyl-CoA oxidations occurred at rates similar to those of valproyl-CoA and octanoyl-CoA oxidase(s), respectively. Studies on fractions obtained from rat liver postnuclear supernatants by isopycnic centrifugation on a linear sucrose density gradient disclose that the density distribution of valproyl-CoA oxidase superimposes to those of catalase, fatty acyl-CoA oxidase and cyanide-insensitive fatty acyl-CoA oxidation, three peroxisomal marker activities. By contrast, the cyanide-insensitive valproyl-CoA oxidation does not adopt the typical peroxisomal distribution of these activities but rather exhibits a mitochondrial localization with, however, a minor peroxisomal component. Interestingly enough, the comparative study of rat tissue distribution, inducibility by clofibrate and sensitivity to deoxycholate indicated that valproyl-CoA oxidase is an enzyme distinct from fatty acyl-CoA oxidase and bile acyl-CoA oxidase. Taken as a whole, the results presented here support the occurrence of a peroxisomal oxidation of the CoA ester of valproic acid and its Δ4-enoic derivate which might be characterized by two major features: initiation by an acyl-CoA oxidase distinct from fatty and bile acyl-CoA oxidases, and inability to complete the β-oxidation cycle which would not proceed, at significant rates, further than the β-hydroxyacyl-CoA dehydrogenation step in peroxisomes.

We describe on a 3-year-old child referred for evaluation and therapy of a cerebral vascular accident with residual hemiplegia and partial epilepsy. Metabolic investigations initially showed normal urinary organic acids as well as normal blood and urinary amino acids. Blood carnitine fractions had been pathological and a secondary carnitine deficiency was diagnosed and treated by oral L-carnitine supplementation. During carnitine treatment, abnormal urinary acylcarnitine profiles were noticed with excessive amounts of several carnitine esters including propionylcarnitine, butyryl-and/or isobutyryl-carnitine, isovaleryl- and/or 2-methylbutyryl-carnitine, hexanoylcarnitine and octanoyl-carnitine. Subsequently, an urinary organic acid profile suggestive of glutaric aciduria type 11 was recorded during a clinical decompensation crisis. Morphological and biochemical studies on skeletal muscle and skin fibroblasts were performed and confirmed the existence of a defect of the mitochondrial β-oxidadation pathways with lipidic myopathy, reduced palmitate and octanoate oxidation rates in cultured fibroblasts. Glutaric aciduria type 11 increases the list of metabolic disorders characterized by hemiplegia and other sequelae of brain ischaemia such as stroke-like episode, seizures, aphasia, ataxia and myoclonia, similar to those seen in MELAS.