Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones

We prove that if ${U\subset \mathbb {R}^n}$ is an open domain whose closure ${\overline U}$ is compact in the path metric, and F is a Lipschitz function on ?U, then for each ${\beta \in \mathbb {R}}$ there exists a unique viscosity solution to the β-biased infinity Laplacian equation $$\beta |\nabla u| + \Delta_\infty u=0$$ on U that extends F, where ${\Delta_\infty u= |\nabla u|^{-2} \sum_{i,j} u_{x_i}u_{x_ix_j} u_{x_j}}$ . In the proof, we extend the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and define the β-biased ${\epsilon}$ -game as follows. The starting position is ${x_0 \in U}$ . At the kth step the two players toss a suitably biased coin (in our key example, player I wins with odds of ${\exp(\beta\epsilon)}$ to 1), and the winner chooses x _k with ${d(x_k,x_{k-1}) < \epsilon}$ . The game ends when ${x_k \in \partial U}$ , and player II pays the amount F(x _k ) to player I. We prove that the value ${u^{\epsilon}(x_0)}$ of this game exists, and that ${\|u^\epsilon - u\|_\infty \to 0}$ as ${\epsilon \to 0}$ , where u is the unique extension of F to ${\overline{U}}$ that satisfies comparison with β-exponential cones. Comparison with exponential cones is a notion that we introduce here, and generalizing a theorem of Crandall, Evans and Gariepy regarding comparison with linear cones, we show that a continuous function satisfies comparison with β-exponential cones if and only if it is a viscosity solution to the β-biased infinity Laplacian equation.     

Synthesis of 4-substituted chlorophthalazines, dihydrobenzoazepinediones, 2-pyrazolylbenzoic acid, and 2-pyrazolylbenzohydrazide via 3-substituted 3-hydroxyisoindolin-1-ones

Herein we describe a general three-step synthesis of 4-substituted chlorophthalazines in good overall yields. In the key step, N,N-dimethylaminophthalimide (8a) directs the selective monoaddition of alkyl, aryl, and heteroaryl organometallic reagents to afford 3-substituted 3-hydroxyisoindolinones 9b, 9i-9am. Many of these hydroxyisoindolinones are converted to chlorophthalazines 1b-1v via reaction with hydrazine, followed by chlorination with POCl(3). We have also discovered two novel transformations of 3-vinyl- and 3-alkynyl-3-hydroxyisoindolinones. Addition of vinyl organometallic reagents to N,N-dimethylaminophthalimide (8a) provided dihydrobenzoazepinediones 15a-15c via the proposed ring expansion of 3-vinyl-3-hydroxyisoindolinone intermediates. 3-Alkynyl-3-hydroxyisoindolinones react with hydrazine and substituted hydrazines to afford 2-pyrazolyl benzoic acids 16a-16d and 2-pyrazolyl benzohydrazides 17a-17g rather than the expected alkynyl phthalazinones.     

Security of quantum bit string commitment depends on the information measure

Unconditionally secure nonrelativistic bit commitment is known to be impossible in both the classical and the quantum world. However, when committing to a string of n bits at once, how far can we stretch the quantum limits? In this Letter, we introduce a framework of quantum schemes where Alice commits a string of n bits to Bob, in such a way that she can only cheat on a bits and Bob can learn at most b bits of information before the reveal phase. Our results are twofold: we show by an explicit construction that in the traditional approach, where the reveal and guess probabilities form the security criteria, no good schemes can exist: a + b is at least n. If, however, we use a more liberal criterion of security, the accessible information, we construct schemes where a = 4log2(n) + O(1) and b = 4, which is impossible classically. Our findings significantly extend known no-go results for quantum bit commitment.     

Macrocyclic Protease Inhibitors with Reduced Peptide Character

There is a real need for simple structures that define a β‐strand conformation, a secondary structure that is central to peptide–protein interactions. For example, protease substrates and inhibitors almost universally adopt this geometry on active site binding. A planar pyrrole is used to replace two amino acids of a peptide backbone to generate a simple macrocycle that retains the required geometry for active site binding. The resulting β‐strand templates have reduced peptide character and provide potent protease inhibitors with the attachment of an appropriate amino aldehyde to the C‐terminus. Picomolar inhibitors of cathepsin L and S are reported and the mode of binding of one example to the model protease chymotrypsin is defined by X‐ray crystallography.     

Contamination of therapeutic human immunoglobulin preparations with apolipoprotein H (β2‐glycoprotein I)

Polyclonal immunoglobulin (Ig) concentrates are important biological medicinal products and the assurance of their quality and safety is crucial. In our present approach we used proteomic methods to check the purity of commercial Ig products of different origin. The experimental setup included nonreducing 2DE or DIGE combined with MALDI‐TOF and the thrombin generation assay, a routine safety test for pharmaceutical Ig preparations, and was complemented by a specific immunoassay. 2DE patterns displayed contaminations with trace amounts of human apolipoprotein H (Apo‐H), transferrin, albumin, and its fragments. In contrast to the latter, Apo‐H is a protein that is active in the coagulation cascade, and thus a potential involvement in thromboembolic events in vivo cannot be excluded. It was found by 2DE and MALDI‐TOF to be a contaminant of several Ig preparations. Spiking experiments of Ig preparations with pure Apo‐H demonstrated an Apo‐H concentration dependent increase in thrombin generation assay values. Traces of Apo‐H are possibly also contributing to unwanted side effects, as already known for factor XIa. The significance of Apo‐H contaminations for these side effects might be verified by detailed analyses of pharmacovigilance data.     

Gas-phase carbene radical anions: new mechanistic insights

The gas-phase reactivity of the CHCl*- anion has been investigated with a series of halomethanes (CCl4, CHCl3, CH2Cl2, and CH3Cl) using a FA-SIFT instrument. Results show that this anion primarily reacts via substitution and by proton transfer. In addition, the reactions of CHCl*- with CHCl3 and CH2Cl2 form minor amounts of Cl2*- and Cl-. The isotopic distribution of these two products is consistent with an insertion-elimination mechanism, where the anion inserts into a C-Cl bond to form an unstable intermediate, which eliminates either Cl2*- or Cl- and Cl*. Neutral and cationic carbenes are known to insert into single bonds; however, this is the first observation of such reactivity for carbene anions.     

Enantioselective Rhodium‐Catalyzed Atom‐Economical Macrolactonization

A highly attractive route toward macrolactones, which form the cyclic scaffold of a multitude of diverse natural compounds, is described. Although many chemical approaches to this structural motif have been explored, an asymmetric variant of the cyclization is unprecedented. Herein we present an enantioselective macrolactonization through an intramolecular atom‐economical rhodium‐catalyzed coupling of ω‐allenyl‐substituted carboxylic acids. The use of a modified diop ligand, chiral DTBM‐diop, led to high enantioselectivity (up to 93 % ee). The reaction tolerated a large variety of functionalities, including α,β‐unsaturated carboxylic acids and depsipeptides, and provided the desired macrocycles with very high enantio‐ and diastereoselectivity.     

One site fits both: A model for the ternary complex of folate + NADPH in R67 dihydrofolate reductase, a D2 symmetric enzyme

R67 dihydrofolate reductase (DHFR) is a novel enzyme that confers resistance to the antibiotic trimethoprim. The crystal structure of R67 DHFR displays a toroidal structure with a central active-site pore. This homotetrameric protein exhibits 222 symmetry, with only a few residues from each chain contributing to the active site, so related sites must be used to bind both substrate (dihydrofolate) and cofactor (NADPH) in the productive R67 DHFR?NADPH?dihydrofolate complex. Whereas the site of folate binding has been partially resolved crystallographically, an interesting question remains: how can the highly symmetrical active site also bind and orient NADPH for catalysis? To model this ternary complex, we employed DOCK and SLIDE, two methods for docking flexible ligands into proteins using quite different algorithms. The bound pteridine ring of folate (Fol I) from the crystal structure of R67 DHFR was used as the basis for docking the nicotinamide-ribose-P_i (NMN) moiety of NADPH. NMN was positioned by both DOCK and SLIDE on the opposite side of the pore from Fol I, where it interacts with Fol I at the pore's center. Numerous residues serve dual roles in binding. For example, Gln 67 from both the B and D subunits has several contacts with the pteridine ring, while the same residue from the A and C subunits has several contacts with the nicotinamide ring. The residues involved in dual roles are generally amphipathic, allowing them to make both hydrophobic and hydrophilic contacts with the ligands. The result is a `hot spot' binding surface allowing the same residues to co-optimize the binding of two ligands, and orient them for catalysis.