Convexity, superquadratic growth, and dot products.

Let P⊂R2 be a point set with cardinality N. We give an improved bound for the number of dot products determined by P, proving that |{p·q:p,q∈P}|≫N2/3+c.A crucial ingredient in the proof of this bound is a new superquadratic expander involving products and shifts. We prove that, for any finite set X⊂R, there exist z,z'∈X such that (zX+1)(2)(z'X+1)(2)(zX+1)(2)(z'X+1)≳|X|5/2.This is derived from a more general result concerning growth of sets defined via convexity and sum sets, and which can be used to prove several other expanders with better than quadratic growth. The proof develops arguments from Hanson, Roche-Newton, and Rudnev [Combinatorica, to appear], and uses predominantly elementary methods.

An ultralong-acting tenofovir ProTide nanoformulation achieves monthslong HBV suppression.

Treatment of chronic hepatitis B virus (HBV) requires lifelong daily therapy. However, suboptimal adherence to the existing daily therapy has led to the need for ultralong-acting antivirals. A lipophilic and hydrophobic ProTide was made by replacing the alanyl isopropyl ester present in tenofovir alafenamide (TAF) with a docosyl phenyl alanyl ester, now referred to as M1TFV. NM1TFV and nanoformulated TAF (NTAF) nanocrystals were formulated by high-pressure homogenization. A single intramuscular injection of NM1TFV, but not NTAF, delivered at a dose of TFV equivalents (168 milligrams per kilogram) demonstrated monthslong antiviral activities in both HBV-transgenic and human hepatocyte transplanted TK-NOG mice. The suppression of HBV DNA in blood was maintained for 3 months. Laboratory experiments on HBV-transfected HepG2.2.15 cells affirmed the animal results and the critical role of docosanol in the sustained NM1TFV antiviral responses. These results provide clear "proof of concept" toward an emerging therapeutic paradigm for the treatment and prevention of HBV infection.

Transformation of dolutegravir into an ultra-long-acting parenteral prodrug formulation.

Ultra-long-acting integrase strand transfer inhibitors were created by screening a library of monomeric and dimeric dolutegravir (DTG) prodrug nanoformulations. This led to an 18-carbon chain modified ester prodrug nanocrystal (coined NM2DTG) with the potential to sustain yearly dosing. Here, we show that the physiochemical and pharmacokinetic (PK) formulation properties facilitate slow drug release from tissue macrophage depot stores at the muscle injection site and adjacent lymphoid tissues following single parenteral injection. Significant plasma drug levels are recorded up to a year following injection. Tissue sites for prodrug hydrolysis are dependent on nanocrystal dissolution and prodrug release, drug-depot volume, perfusion, and cell-tissue pH. Each affect an extended NM2DTG apparent half-life recorded by PK parameters. The NM2DTG product can impact therapeutic adherence, tolerability, and access of a widely used integrase inhibitor in both resource limited and rich settings to reduce HIV-1 transmission and achieve optimal treatment outcomes.