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This article describes a formal proof of the Kepler conjecture on dense sphere packings in a combination of the HOL Light and Isabelle proof assistants. This paper constitutes the official published account of the now completed Flyspeck project.

The telomerase complex is responsible for telomere maintenance and represents a promising cancer therapeutic target. We describe herein the antitelomerase and antitumor properties of a small-molecule compound designed by computer modeling to interact with and stabilize human G-quadruplex DNA, a structure that may form with telomeric DNA, thereby inhibiting access to telomerase. The 3,6,9-trisubstituted acridine 9-[4-(N,N-dimethylamino)phenylamino]-3,6-bis(3-pyrrolodinopropionamido) acridine (BRACO19) represents one of the most potent cell-free inhibitors of human telomerase yet described (50% inhibitory concentration of 115 Ϯ 18 nM). Moreover, in contrast to G-quadruplex interactive agents described previously, BRACO19 did not cause nonspecific acute cytotoxicity at similar concentrations to those required to completely inhibit telomerase activity. There exists a 90-fold differential (mean 50% inhibitory concentration for acute cell kill across seven human tumor cell lines of 10.6 Ϯ 0.7 M). The exposure of 21NT human breast cancer cells, which possess relatively short telomeres, to nonacute cytotoxic concentrations of BRACO19 (2 M) resulted in a marked reduction in cell growth after only 15 days. This was concomitant with a reduction in intracellular telomerase activity and onset of senescence as indicated by an increase in the number of ␤-galactosidase positive-staining cells. Intraperitoneal administration of nontoxic doses of BRACO19 (2 mg/kg) to mice bearing advanced stage A431 human vulval carcinoma subcutaneous xenografts and previously treated with paclitaxel induced a significant increase in antitumor effect compared with that observed with paclitaxel alone. BRACO19 thus represents the first of a "second generation" of G-quadruplex-mediated telomerase/telomere-interactive compounds. It possesses nanomolar potency against telomerase but low nonspecific cytotoxicity, growth inhibitory effects, and induction of senescence in a human breast cancer cell line and, moreover, significant antitumor activity in vivo when administered post paclitaxel to mice bearing a human tumor xenograft carcinoma.

Inhibition of the ability of the enzyme telomerase to add telomeric repeats to the end of chromosomes is a novel target for potential anticancer therapy. This paper examines the hypothesis that compounds possessing a planar aromatic chromophore inhibit telomerase via stabilization of, and binding to, a folded guanine quadruplex structure. Two series of telomerase inhibitors have been designed based on the 2,6-disubstituted amidoanthracene-9,10-dione and 3,6-disubstituted acridine chromophores in order to investigate structure-activity relationships between biological activity and substituent group size. The relative binding energies between these compounds and the folded human telomere DNA quadruplex were determined using molecular simulation methods, involving explicitly solvated structures. The results obtained are in excellent agreement with the biological activity as measured in vitro using a modified TRAP assay and in general agreement with the ranking order of binding enthalpies found in isothermal titration calorimetry studies. This broad agreement provides strong support for the hypothesis that guanine quadruplexes are the primary target for telomerase inhibitors with extended planar chromophores.