In this paper we introduce a definition of the pre-Schwarzian and the Schwarzian derivatives of any locally univalent harmonic mapping f in the complex plane without assuming any additional condition on the (second complex) dilatation ω f of f . Using the new definition for the Schwarzian derivative of harmonic mappings, we prove analogous theorems to those by Chuaqui, Duren, and Osgood. Also, we obtain a Becker-type criterion for the univalence of harmonic mappings.

In many prokaryotes, a specific DNA primase/polymerase (PolDom) is required for nonhomologous end joining (NHEJ) repair of DNA double-strand breaks (DSBs). Here, we report the crystal structure of a catalytically active conformation of Mycobacterium tuberculosis PolDom, consisting of a polymerase bound to a DNA end with a 3' overhang, two metal ions, and an incoming nucleotide but, significantly, lacking a primer strand. This structure represents a polymerase:DNA complex in a preternary intermediate state. This polymerase complex occurs in solution, stabilizing the enzyme on DNA ends and promoting nucleotide extension of short incoming termini. We also demonstrate that the invariant Arg(220), contained in a conserved loop (loop 2), plays an essential role in catalysis by regulating binding of a second metal ion in the active site. We propose that this NHEJ intermediate facilitates extension reactions involving critically short or noncomplementary DNA ends, thus promoting break repair and minimizing sequence loss during DSB repair.

DNA polymerase mu (Pol) is a family X member implicated in DNA repair, with template-directed and terminal transferase (templateindependent) activities. It has been proposed that the terminal transferase activity of Pol can be specifically required during non-homologous end joining (NHEJ) to create or increase complementarity of DNA ends. By site-directed mutagenesis in human Pol, we have identified a specific DNA ligand residue (Arg 387 ) that is responsible for its limited terminal transferase activity compared to that of human TdT, its closest homologue (42% amino acid identity). Pol mutant R387K (mimicking TdT) displayed a large increase in terminal transferase activity, but a weakened interaction with ssDNA. That paradox can be explained by the regulatory role of Arg 387 in the translocation of the primer from a nonproductive E:DNA complex to a productive E:DNA:dNTP complex in the absence of a templating base, which is probably the rate limiting step during template-independent synthesis. Further, we show that the Pol switch from terminal transferase to templated insertions in NHEJ reactions is triggered by recognition of a 5-P at a second DNA end, whose 3-protrusion could provide a templating base to facilitate such a special ''pre-catalytic translocation step.'' These studies shed light on the mechanism by which a rate-limited terminal transferase activity in Pol could regulate the balance between accuracy and necessary efficiency, providing some variability during NHEJ. P hysical and chemical damage of DNA is the root cause of a large number of human syndromes, including premature aging, various cancer predispositions, and congenital abnormalities. To overcome DNA damage and maintain stability in the chromosomes, cells contain an array of specific DNA repair pathways that act upon the different kinds of lesions (1). Specialized DNA polymerases are essential actors within these pathways and, in humans, at least 12 are devoted to overcome or repair DNA damage in the cell (2). DNA polymerases of the X family, which in mammals include DNA polymerases beta (Polß), lambda (Pol), mu (Pol), and terminal deoxynucleotidyl transferase (TdT), are structurally related enzymes specialized in repair pathways involving double strand breaks (DSB) and gaps (3).Unlike Polß, Pol, Pol, and TdT contain a BRCA-1 Cterminal (BRCT) interaction domain at their N-termini (4-7), that serves to interact with the core NHEJ factors Ku and XRCC4-Ligase IV (8-12), suggesting their involvement in DSB repair via NHEJ. Whereas DSB repair by homologous recombination is highly accurate because it relies directly on complementary DNA sequences, the accuracy of the repair reaction by NHEJ is lower, somehow compromising preservation of the original DNA information. In this context, NHEJ DNA polymerases would be needed either to extend the 3Ј-end at 5Ј-overhangs, to fill gaps created by limited complementarity of the two DNA ends, or eventually in the case of joining noncomplementary (incompatible) ends, to add sequences at the 3Ј-end to create a...