Noether-type discrete conserved quantities arising from a finite element approximation of a variational problem

In this work, we prove a weak Noether-type Theorem for a class of variational problems that admit broken extremals. We use this result to prove discrete Noether-type conservation laws for a conforming finite element discretisation of a model elliptic problem. In addition, we study how well the finite element scheme satisfies the continuous conservation laws arising from the application of Noether’s first theorem (1918). We summarise extensive numerical tests, illustrating the conservation of the discrete Noether law using the p-Laplacian as an example and derive a geometric-based adaptive algorithm where an appropriate Noether quantity is the goal functional.

Production of bioactive diterpenoids in the Euphorbiaceae depends on evolutionarily conserved gene clusters

The Euphorbiaceae produce a diverse range of diterpenoids, many of which have pharmacological activities. These diterpenoids include ingenol mebutate, which is licensed for the treatment of a precancerous skin condition (actinic keratosis), and phorbol derivatives such as resiniferatoxin and prostratin, which are undergoing investigation for the treatment of severe pain and HIV, respectively. Despite the interest in these diterpenoids, their biosynthesis is poorly understood at present, with the only characterized step being the conversion of geranylgeranyl pyrophosphate into casbene. Here, we report a physical cluster of diterpenoid biosynthetic genes from castor (Ricinus communis), including casbene synthases and cytochrome P450s from the CYP726A subfamily. CYP726A14, CYP726A17, and CYP726A18 were able to catalyze 5-oxidation of casbene, a conserved oxidation step in the biosynthesis of this family of medicinally important diterpenoids. CYP726A16 catalyzed 7,8-epoxidation of 5-keto-casbene and CYP726A15 catalyzed 5-oxidation of neocembrene. Evidence of similar gene clustering was also found in two other Euphorbiaceae, including Euphorbia peplus, the source organism of ingenol mebutate. These results demonstrate conservation of gene clusters at the higher taxonomic level of the plant family and that this phenomenon could prove useful in further elucidating diterpenoid biosynthetic pathways.