No abstract

A construction analogous to that of Godefroy-Kalton for metric spaces allows one to embed isometrically, in a canonical way, every quasi-metric space (X, d) in an asymmetric normed space Fa(X, d) (its quasi-metric free space, also called asymmetric free space or semi-Lipschitz free space). The quasi-metric free space satisfies a universal property (linearization of semi-Lipschitz functions). The (conic) dual of Fa(X, d) coincides with the non-linear asymmetric dual of (X, d), that is, the space SLip 0 (X, d) of semi-Lipschitz functions on (X, d), vanishing at a base point. In particular, for the case of a metric space (X, D), the above construction yields its usual free space. On the other hand, every metric space (X, D) naturally inherits a canonical asymmetrization coming from its free space F(X). This gives rise to a quasi-metric space (X, D+) and an asymmetric free space Fa(X, D+). The symmetrization of the latter is isomorphic to the original free space F(X). The results of this work are illustrated with explicit examples.

Xanthophyllomyces dendrorhous synthesizes astaxanthin, a carotenoid used in aquaculture. Astaxanthin is synthesized from metabolites of the mevalonate pathway, which are also precursors for sterols biosynthesis. The interruption of the CYP61 gene, which is involved in the synthesis of ergosterol (mutant CBS.cyp61 − ), resulted in a phenotype that overproduces carotenoids due to the activation of the SREBP pathway. In this work, we constructed other mutants of ergosterol biosynthesis in this yeast to evaluate whether they have the same phenotype as mutant CBS.cyp61 − . By bioinformatic analysis, the ERG3 and ERG4 genes of X. dendrorhous were identified, and each gene was deleted in the wild-type strain. Mutants CBS. erg3 and CBS. erg4 did not produce ergosterol; CBS. erg3 primarily accumulated episterol, and CBS. erg4 primarily accumulated ergosta-5,7,22,24(28)-tetraenol. The transcription levels of the HMGS gene of the mevalonate pathway were evaluated by RT-qPCR, which showed a slight increase in CBS. erg4, but the transcription levels were still 10-fold lower than in strain CBS.cyp61 − . Both CBS. erg3 and CBS. erg4 did not overproduce carotenoids, even though they do not produce ergosterol. Thus, the results of this study indicate that the absence of ergosterol does not activate the SREBP pathway in X. dendrorhous, but rather it depends on other alterations in sterol composition.

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A construction analogous to that of Godefroy-Kalton for metric spaces allows to embed isometrically, in a canonical way, every quasi-metric space (X, d) to an asymmetric normed space Fa(X, d) (its quasi-metric free space, also called asymmetric free space or semi-Lipschitz free space). The quasimetric free space satisfies a universal property (linearization of semi-Lipschitz functions). The (conic) dual of Fa(X, d) coincides with the nonlinear asymmetric dual of (X, d), that is, the space SLip 0 (X, d) of semi-Lipschitz functions on (X, d), vanishing at a base point. In particular, for the case of a metric space (X, D), the above construction yields its usual free space. On the other hand, every metric space (X, D) inherits naturally a canonical asymmetrization coming from its free space F (X). This gives rise to a quasi-metric space (X, D + ) and an asymmetric free space Fa(X, D + ). The symmetrization of the latter is isomorphic to the original free space F (X). The results of this work are illustrated with explicit examples.