The concept of “character” in Dirichlet’s theorem on primes in an arithmetic progression

In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. We survey implicit and explicit uses of Dirichlet characters in presentations of Dirichlet’s proof in the nineteenth and early twentieth centuries, with an eye toward understanding some of the pragmatic pressures that shaped the evolution of modern mathematical method.     

Small molecule activators of SIRT1 as therapeutics for the treatment of type 2 diabetes

Calorie restriction extends lifespan and produces a metabolic profile desirable for treating diseases of ageing such as type 2 diabetes. SIRT1, an NAD+-dependent deacetylase, is a principal modulator of pathways downstream of calorie restriction that produce beneficial effects on glucose homeostasis and insulin sensitivity. Resveratrol, a polyphenolic SIRT1 activator, mimics the anti-ageing effects of calorie restriction in lower organisms and in mice fed a high-fat diet ameliorates insulin resistance, increases mitochondrial content, and prolongs survival. Here we describe the identification and characterization of small molecule activators of SIRT1 that are structurally unrelated to, and 1,000-fold more potent than, resveratrol. These compounds bind to the SIRT1 enzyme-peptide substrate complex at an allosteric site amino-terminal to the catalytic domain and lower the Michaelis constant for acetylated substrates. In diet-induced obese and genetically obese mice, these compounds improve insulin sensitivity, lower plasma glucose, and increase mitochondrial capacity. In Zucker fa/fa rats, hyperinsulinaemic-euglycaemic clamp studies demonstrate that SIRT1 activators improve whole-body glucose homeostasis and insulin sensitivity in adipose tissue, skeletal muscle and liver. Thus, SIRT1 activation is a promising new therapeutic approach for treating diseases of ageing such as type 2 diabetes.     

Adaptive immunity maintains occult cancer in an equilibrium state

The capacity of immunity to control and shape cancer, that is, cancer immunoediting, is the result of three processes that function either independently or in sequence: elimination (cancer immunosurveillance, in which immunity functions as an extrinsic tumour suppressor in naive hosts); equilibrium (expansion of transformed cells is held in check by immunity); and escape (tumour cell variants with dampened immunogenicity or the capacity to attenuate immune responses grow into clinically apparent cancers). Extensive experimental support now exists for the elimination and escape processes because immunodeficient mice develop more carcinogen-induced and spontaneous cancers than wild-type mice, and tumour cells from immunodeficient mice are more immunogenic than those from immunocompetent mice. In contrast, the equilibrium process was inferred largely from clinical observations, including reports of transplantation of undetected (occult) cancer from organ donor into immunosuppressed recipients. Herein we use a mouse model of primary chemical carcinogenesis and demonstrate that equilibrium occurs, is mechanistically distinguishable from elimination and escape, and that neoplastic cells in equilibrium are transformed but proliferate poorly in vivo. We also show that tumour cells in equilibrium are unedited but become edited when they spontaneously escape immune control and grow into clinically apparent tumours. These results reveal that, in addition to destroying tumour cells and sculpting tumour immunogenicity, the immune system of a naive mouse can also restrain cancer growth for extended time periods.     

A second generation human haplotype map of over 3.1 million SNPs

We describe the Phase II HapMap, which characterizes over 3.1 million human single nucleotide polymorphisms (SNPs) genotyped in 270 individuals from four geographically diverse populations and includes 25-35% of common SNP variation in the populations surveyed. The map is estimated to capture untyped common variation with an average maximum r2 of between 0.9 and 0.96 depending on population. We demonstrate that the current generation of commercial genome-wide genotyping products captures common Phase II SNPs with an average maximum r2 of up to 0.8 in African and up to 0.95 in non-African populations, and that potential gains in power in association studies can be obtained through imputation. These data also reveal novel aspects of the structure of linkage disequilibrium. We show that 10-30% of pairs of individuals within a population share at least one region of extended genetic identity arising from recent ancestry and that up to 1% of all common variants are untaggable, primarily because they lie within recombination hotspots. We show that recombination rates vary systematically around genes and between genes of different function. Finally, we demonstrate increased differentiation at non-synonymous, compared to synonymous, SNPs, resulting from systematic differences in the strength or efficacy of natural selection between populations.     

Genome-wide detection and characterization of positive selection in human populations

With the advent of dense maps of human genetic variation, it is now possible to detect positive natural selection across the human genome. Here we report an analysis of over 3 million polymorphisms from the International HapMap Project Phase 2 (HapMap2). We used 'long-range haplotype' methods, which were developed to identify alleles segregating in a population that have undergone recent selection, and we also developed new methods that are based on cross-population comparisons to discover alleles that have swept to near-fixation within a population. The analysis reveals more than 300 strong candidate regions. Focusing on the strongest 22 regions, we develop a heuristic for scrutinizing these regions to identify candidate targets of selection. In a complementary analysis, we identify 26 non-synonymous, coding, single nucleotide polymorphisms showing regional evidence of positive selection. Examination of these candidates highlights three cases in which two genes in a common biological process have apparently undergone positive selection in the same population:LARGE and DMD, both related to infection by the Lassa virus, in West Africa;SLC24A5 and SLC45A2, both involved in skin pigmentation, in Europe; and EDAR and EDA2R, both involved in development of hair follicles, in Asia.     

An integrated semiconductor device enabling non-optical genome sequencing

The seminal importance of DNA sequencing to the life sciences, biotechnology and medicine has driven the search for more scalable and lower-cost solutions. Here we describe a DNA sequencing technology in which scalable, low-cost semiconductor manufacturing techniques are used to make an integrated circuit able to directly perform non-optical DNA sequencing of genomes. Sequence data are obtained by directly sensing the ions produced by template-directed DNA polymerase synthesis using all-natural nucleotides on this massively parallel semiconductor-sensing device or ion chip. The ion chip contains ion-sensitive, field-effect transistor-based sensors in perfect register with 1.2 million wells, which provide confinement and allow parallel, simultaneous detection of independent sequencing reactions. Use of the most widely used technology for constructing integrated circuits, the complementary metal-oxide semiconductor (CMOS) process, allows for low-cost, large-scale production and scaling of the device to higher densities and larger array sizes. We show the performance of the system by sequencing three bacterial genomes, its robustness and scalability by producing ion chips with up to 10 times as many sensors and sequencing a human genome.     

Two ten-billion-solar-mass black holes at the centres of giant elliptical galaxies

Observational work conducted over the past few decades indicates that all massive galaxies have supermassive black holes at their centres. Although the luminosities and brightness fluctuations of quasars in the early Universe suggest that some were powered by black holes with masses greater than 10 billion solar masses, the remnants of these objects have not been found in the nearby Universe. The giant elliptical galaxy Messier 87 hosts the hitherto most massive known black hole, which has a mass of 6.3 billion solar masses. Here we report that NGC 3842, the brightest galaxy in a cluster at a distance from Earth of 98 megaparsecs, has a central black hole with a mass of 9.7 billion solar masses, and that a black hole of comparable or greater mass is present in NGC 4889, the brightest galaxy in the Coma cluster (at a distance of 103 megaparsecs). These two black holes are significantly more massive than predicted by linearly extrapolating the widely used correlations between black-hole mass and the stellar velocity dispersion or bulge luminosity of the host galaxy. Although these correlations remain useful for predicting black-hole masses in less massive elliptical galaxies, our measurements suggest that different evolutionary processes influence the growth of the largest galaxies and their black holes.     

Overcoming lability of extremely long alkane carbon-carbon bonds through dispersion forces

Steric effects in chemistry are a consequence of the space required to accommodate the atoms and groups within a molecule, and are often thought to be dominated by repulsive forces arising from overlapping electron densities (Pauli repulsion). An appreciation of attractive interactions such as van der Waals forces (which include London dispersion forces) is necessary to understand chemical bonding and reactivity fully. This is evident from, for example, the strongly debated origin of the higher stability of branched alkanes relative to linear alkanes and the possibility of constructing hydrocarbons with extraordinarily long C-C single bonds through steric crowding. Although empirical bond distance/bond strength relationships have been established for C-C bonds (longer C-C bonds have smaller bond dissociation energies), these have no present theoretical basis. Nevertheless, these empirical considerations are fundamental to structural and energetic evaluations in chemistry, as summarized by Pauling as early as 1960 and confirmed more recently. Here we report the preparation of hydrocarbons with extremely long C-C bonds (up to 1.704??), the longest such bonds observed so far in alkanes. The prepared compounds are unexpectedly stable--noticeable decomposition occurs only above 200?°C. We prepared the alkanes by coupling nanometre-sized, diamond-like, highly rigid structures known as diamondoids. The extraordinary stability of the coupling products is due to overall attractive dispersion interactions between the intramolecular H???H contact surfaces, as is evident from density functional theory computations with and without inclusion of dispersion corrections.