A Fresh Variational-Analysis Look at the Positive Semidefinite Matrices World

International audienceEngineering sciences and applications of mathematics show unambiguously that positive semidefiniteness of matrices is the most important generalization of non-negative real num- bers. This notion of non-negativity for matrices has been well-studied in the literature; it has been the subject of review papers and entire chapters of books. This paper reviews some of the nice, useful properties of positive (semi)definite matrices, and insists in particular on (i) characterizations of positive (semi)definiteness and (ii) the geometrical properties of the set of positive semidefinite matrices. Some properties that turn out to be less well-known have here a special treatment. The use of these properties in optimization, as well as various references to applications, are spread all the way through. The "raison d'être" of this paper is essentially pedagogical; it adopts the viewpoint of variational analysis, shedding new light on the topic. Important, fruitful, and subtle, the positive semidefinite world is a good place to start with this domain of applied mathematics

UVSSA and USP7, a new couple in transcription-coupled DNA repair

Transcription-coupled nucleotide excision repair (TC-NER) specifically removes transcription-blocking lesions from our genome. Defects in this pathway are associated with two human disorders: Cockayne syndrome (CS) and UV-sensitive syndrome (UVSS). Despite a similar cellular defect in the UV DNA damage response, patients with these syndromes exhibit strikingly distinct symptoms; CS patients display severe developmental, neurological, and premature aging features, whereas the phenotype of UVSS patients is mostly restricted to UV hypersensitivity. The exact molecular mechanism behind these clinical differences is still unknown; however, they might be explained by additional functions of CS proteins beyond TC-NER. A short overview of the current hypotheses addressing possible molecular mechanisms and the proteins involved are presented in this review. In addition, we will focus on two new players involved in TC-NER which were recently identified: UV-stimulated scaffold protein A (UVSSA) and ubiquitin-specific protease 7 (USP7). UVSSA has been found to be the causative gene for UVSS and, together with USP7, is implicated in regulating TC-NER activity. We will discuss the function of UVSSA and USP7 and how the discovery of these proteins contributes to a better understanding of the molecular mechanisms underlying the clinical differences between UVSS and the more severe CS

Elongation factor ELOF1 drives transcription-coupled repair and prevents genome instability

Correct transcription is crucial for life. However, DNA damage severely impedes elongating RNA polymerase II, causing transcription inhibition and transcription-replication conflicts. Cells are equipped with intricate mechanisms to counteract the severe consequence of these transcription-blocking lesions. However, the exact mechanism and factors involved remain largely unknown. Here, using a genome-wide CRISPR-Cas9 screen, we identified the elongation factor ELOF1 as an important factor in the transcription stress response following DNA damage. We show that ELOF1 has an evolutionarily conserved role in transcription-coupled nucleotide excision repair (TC-NER), where it promotes recruitment of the TC-NER factors UVSSA and TFIIH to efficiently repair transcription-blocking lesions and resume transcription. Additionally, ELOF1 modulates transcription to protect cells against transcription-mediated replication stress, thereby preserving genome stability. Thus, ELOF1 protects the transcription machinery from DNA damage via two distinct mechanisms