Coloring finite subsets of uncountable sets

It is consistent for every that and there is a function such that every finite set can be written in at most ways as the union of two distinct monocolored sets. If GCH holds, for every such coloring there is a finite set that can be written at least ways as the union of two sets with the same color.

     

A domination algorithm for {0,1}‐instances of the travelling salesman problem

We present an approximation algorithm for ‐instances of the travelling salesman problem which performs well with respect to combinatorial dominance. More precisely, we give a polynomial‐time algorithm which has domination ratio . In other words, given a ‐edge‐weighting of the complete graph on vertices, our algorithm outputs a Hamilton cycle of with the following property: the proportion of Hamilton cycles of whose weight is smaller than that of is at most . Our analysis is based on a martingale approach. Previously, the best result in this direction was a polynomial‐time algorithm with domination ratio for arbitrary edge‐weights. We also prove a hardness result showing that, if the Exponential Time Hypothesis holds, there exists a constant such that cannot be replaced by in the result above. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 427–453, 2016     

In‐Bead Screening of Hydroxamic Acids for the Identification of HDAC Inhibitors

A one bead–one compound screening format is presented. Following solid‐phase synthesis on a photolabile linker, library compounds were readily released and screened inside polymer beads. The release of screening compounds was readily controlled by varying photolysis time and light intensity. Dose‐response experiments were carried out to effectively distinguish high‐ and low‐affinity ligands. A library containing 55 800 compounds was synthesized and screened in a fluorometric assay, thereby identifying potent HDAC inhibitors with IC₅₀ values in the nanomolar range.     

Fragment Linking and Optimization of Inhibitors of the Aspartic Protease Endothiapepsin: Fragment‐Based Drug Design Facilitated by Dynamic Combinatorial Chemistry

Fragment‐based drug design (FBDD) affords active compounds for biological targets. While there are numerous reports on FBDD by fragment growing/optimization, fragment linking has rarely been reported. Dynamic combinatorial chemistry (DCC) has become a powerful hit‐identification strategy for biological targets. We report the synergistic combination of fragment linking and DCC to identify inhibitors of the aspartic protease endothiapepsin. Based on X‐ray crystal structures of endothiapepsin in complex with fragments, we designed a library of bis‐acylhydrazones and used DCC to identify potent inhibitors. The most potent inhibitor exhibits an IC₅₀ value of 54 nm , which represents a 240‐fold improvement in potency compared to the parent hits. Subsequent X‐ray crystallography validated the predicted binding mode, thus demonstrating the efficiency of the combination of fragment linking and DCC as a hit‐identification strategy. This approach could be applied to a range of biological targets, and holds the potential to facilitate hit‐to‐lead optimization.     

一类最优组合批处理码

Ishai等人首先提出了批处理码的概念,Peterson等人从纯组合的观点定义了(n,N,k,m)-组合批处理码:即是一个n元集和它的m个子集组成的集合系统,对于整数尼,满足任意k个元素都能从每个子集中至多读取1个元素(可以一般化为t个元素)来取得,此时m个子集中元素的总数为N.对给定的参数n,k,m,确定N的最小值N(n,k,m)是该问题研究的中心内容,它不仅具有理论意义,而且有着重要的使用价值.到目前为止,除了一些极特殊的参数以外,当k≥5,m+3≤n(m k-2)时,N(n,k,m)的值还没有被确定.本文给出了N(m+3,5,m)=m+11(m≥7),N(9,5,6)=18,N(m+3,6,m)=m+13(m≥8),N(10,6,7)=21.得到的结果部分解决了:Peterson等人提出的未解决问题.     

Maximal Induced Matchings in Triangle‐Free Graphs

An induced matching in a graph is a set of edges whose endpoints induce a 1‐regular subgraph. It is known that every n‐vertex graph has at most  maximal induced matchings, and this bound is the best possible. We prove that every n‐vertex triangle‐free graph has at most  maximal induced matchings; this bound is attained by every disjoint union of copies of the complete bipartite graph K_{3, 3}. Our result implies that all maximal induced matchings in an n‐vertex triangle‐free graph can be listed in time , yielding the fastest known algorithm for finding a maximum induced matching in a triangle‐free graph.     

Solid-phase synthesis of tetrahydro-1,4-benzodiazepin-2-one derivatives

An efficient method for solid-phase construction of tetrahydro-1,4-benzodiazepin-2-one scaffold is described. Polymer-bound 4-(bromomethyl)-3-nitrobenzoic acid was reacted with alpha-amino acid methyl esters, followed by nitro group reduction and hydrolysis. Subsequent intramolecular cyclization and alkylation at N(4) afforded the structurally diverse products in high yields and excellent purities.     

Identification of selective inhibitors of acetylcholinesterase from a combinatorial library of 2,5-piperazinediones

The potentiation of central cholinergic activity has been proposed as a therapeutic approach for improving cognitive function in patients with Alzheimer's disease. Increasing the acetylcholine concentration in brain by modulating acetylcholinesterase (AChE) activity is among the most promising strategies. We have used a combinatorial approach to identify different 2,5-piperazinediones (DKP) with AChE inhibitory activity. Our goal was to find inhibitors exhibiting high AChE/BuChE (butyrylcholinesterase) selectivity, in order to reduce the undesirable side effects elicited by most of the inhibitors that have been developed to date. Screening of a DKP library constructed on solid-phase using the multiple parallel synthesis format, resulted in the identification of several compounds with moderate efficacy on AChE. In particular, DKP-80 had an IC50 = 2.2 microM with no significant inhibitory activity on BuChE. Moreover, estimated values of Clog P and log BB for the most active compounds fulfilled the bioavailability requirements for enzyme inhibitors acting on the central nervous system. In order to understand the inhibitory properties of the ligand at the molecular level, molecular dynamics simulations were computed on DKP-80 complexed to AChE, and the most relevant binding interactions of this inhibitor to the active center of the enzyme were characterized. Overall the present results indicate that the DKP-based compounds identified are novel AChE inhibitors which may be considered likely lead compounds for further development of drug candidates against Alzheimer's disease.     

Single bead parallel synthesis and screening

Microstructured silicon wafers were employed as miniaturized solid-phase reaction vessels as well as miniaturized micro titer plates. Employing piezoelectric drop-on-demand liquid jets, a combinatorial library of 256 Peptides was synthesized on single beads. The synthesis protocol was associated to the location in the silicon nano-well arrangement. Products were photolytically cleaved in the same well that was used for synthesis and subsequently interrogated for thrombin inhibition in a homogeneous competition assay. The assay procedure was based on drop-on-demand liquid delivery and laser induced fluorescence imaging. The novel format proved useful for the integration of both synthesis and screening into one platform, a prerequisite for an iterative, evolutionary approach towards drug discovery.     

Regioselective alkylation at the N4 positionof a 3-oxo-1,4-benzodiazepine on solid support

An efficient solid phase regioselective alkylation at the N4 positionof a 3-oxo-1,4-benzodiazepine template exemplified by 4-H-2,3,4,5-tetrahydro-7-iodo-3-oxo-1H-1,4-benzodiazepine-2-acetate-polymerester is described. Further chemical elaboration atposition 7, utilizing a modified Heck reaction, allows the incorporationof amides from primary or secondary amines. The two diversity points atpositions 4 and 7 were utilized tosynthesize a 28-membered, combinatorial array on Sasrin® resin in moderate yields and >80% purity. Having validated the chemistry on solid support, acombine and splitapproach to prepare a bead-bound combinatorial library is achievableutilizing similar experimental practices and procedures as in thearray synthesis.