Prime numbers are the multiplicative building blocks of natural numbers. Understanding their overall influence and especially their distribution gives rise to central questions in mathematics and physics. In particular their finer distribution is closely connected with the Riemann hypothesis, the most important unsolved problem in the mathematical world. Assuming only subjects covered in a standard degree in mathematics, the authors comprehensively cover all the topics met in first courses on multiplicative number theory and the distribution of prime numbers. They bring their extensive and distinguished research expertise to bear in preparing the student for intelligent reading of the more advanced research literature. This 2006 text, which is based on courses taught successfully over many years at Michigan, Imperial College and Pennsylvania State, is enriched by comprehensive historical notes and references as well as over 500 exercises.

The recent outbreak of Zika virus (ZIKV) has infected over 1 million people in over 30 countries. ZIKV replicates its RNA genome using virally encoded replication proteins. Nonstructural protein 5 (NS5) contains a methyltransferase for RNA capping and a polymerase for viral RNA synthesis. Here we report the crystal structures of full-length NS5 and its polymerase domain at 3.0 Å resolution. The NS5 structure has striking similarities to the NS5 protein of the related Japanese encephalitis virus. The methyltransferase contains in-line pockets for substrate binding and the active site. Key residues in the polymerase are located in similar positions to those of the initiation complex for the hepatitis C virus polymerase. The polymerase conformation is affected by the methyltransferase, which enables a more efficiently elongation of RNA synthesis in vitro. Overall, our results will contribute to future studies on ZIKV infection and the development of inhibitors of ZIKV replication.

Using cross-linking coupled to matrix-assisted laser desorption/ionization mass spectrometry and CLIP-Seq sequencing, we determined the peptide and oligonucleotide sequences at the interfaces between the capsid proteins and the genomic RNA of bacteriophage MS2. The results suggest that the same coat protein (CP)–RNA and maturation protein (MP)–RNA interfaces are used in every viral particle. The portions of the viral RNA in contact with CP subunits span the genome, consistent with a large number of discrete and similar contacts within each particle. Many of these sites match previous predictions of the locations of multiple, dispersed and degenerate RNA sites with cognate CP affinity termed packaging signals (PSs). Chemical RNA footprinting was used to compare the secondary structures of protein-free genomic fragments and the RNA in the virion. Some PSs are partially present in protein-free RNA but others would need to refold from their dominant solution conformations to form the contacts identified in the virion. The RNA-binding peptides within the MP map to two sections of the N-terminal half of the protein. Comparison of MP sequences from related phages suggests a similar arrangement of RNA-binding sites, although these N-terminal regions have only limited sequence conservation. In contrast, the sequences of the C-termini are highly conserved, consistent with them encompassing pilin-binding domains required for initial contact with host cells. These results provide independent and unambiguous support for the assembly of MS2 virions via a PS-mediated mechanism involving a series of induced-fit viral protein interactions with RNA.

No abstract

Protein therapeutics are rapidly transforming the pharmaceutical industry. Unlike for small molecule therapeutics, current technologies are challenged to provide the rapid, high resolution analyses of protein higher order structures needed to ensure drug efficacy and safety. Consequently, significant attention has turned to developing new methods that can quickly, accurately, and reproducibly characterize the three-dimensional structure of protein therapeutics. In this work, we describe a method that uses diethylpyrocarbonate (DEPC) labeling and mass spectrometry to detect three-dimensional structural changes in therapeutic proteins that have been exposed to degrading conditions. Using β2-microglobulin, immunoglobulin G1, and human growth hormone as model systems, we demonstrate that DEPC labeling can identify both specific protein regions that mediate aggregation and those regions that undergo more subtle structural changes upon mishandling of these proteins. Importantly, DEPC labeling is able to provide information for up to 30% of the surface residues in a given protein, thereby providing excellent structural resolution. Given the simplicity of the DEPC labeling chemistry and the relatively straightforward mass spectral analysis of DEPC-labeled proteins, we expect this method should be amenable to a wide range of protein therapeutics and their different formulations.