Analogues of Ramanujan’s partition identities

Ramanujan discovered that $$\sum_{n=0}^\infty p(5n+4)q^n=5 \prod_{j=1}^\infty \frac{(1-q^{5j})^5}{(1-q^j)^6}, $$ where p(n) is the number of partitions of n. Recently, H.-C. Chan and S. Cooper, and H.H. Chan and P.C. Toh established several analogues of Ramanujan’s partition identities by employing the theory of modular functions. Very recently, N.D. Baruah and K.K. Ojah studied the partition function $p_{[c^{l}d^{m}]}(n)$ which is defined by $$\sum_{n=0}^\infty p_{[c^ld^m]}(n)q^n= \frac{1}{\prod_{j=1}^\infty (1-q^{cj})^{l}(1-q^{dj})^m}. $$ They discovered some analogues of Ramanujan’s partition identities and deduced several interesting partition congruences. In this paper, we provide a uniform method to prove some of their results by utilizing an addition formula. In the process, we also establish some new analogues of Ramanujan’s partition identities and congruences for $p_{[c^{l}d^{m}]}(n)$ .     

PEG–Anthracene Hydrogels as an On‐Demand Stiffening Matrix To Study Mechanobiology

There is a growing interest in materials that can dynamically change their properties in the presence of cells to study mechanobiology. Herein, we exploit the 365 nm light mediated [4+4] photodimerization of anthracene groups to develop cytocompatible PEG‐based hydrogels with tailorable initial moduli that can be further stiffened. A hydrogel formulation that can stiffen from 10 to 50 kPa, corresponding to the stiffness of a healthy and fibrotic heart, respectively, was prepared. This system was used to monitor the stiffness‐dependent localization of NFAT, a downstream target of intracellular calcium signaling using a reporter in live cardiac fibroblasts (CFbs). NFAT translocates to the nucleus of CFbs on stiffening hydrogels within 6 h, whereas it remains cytoplasmic when the CFbs are cultured on either 10 or 50 kPa static hydrogels. This finding demonstrates how dynamic changes in the mechanical properties of a material can reveal the kinetics of mechanoresponsive cell signaling pathways that may otherwise be missed in cells cultured on static substrates.     

Selective hetero-trisfunctionalization of the large rim of a biomimetic calix[6]arene using host-guest chemistry as a synthetic tool

A calix[6]arene has been selectively functionalized by three different groups at the large rim. The strategy relies on the hostguest recognition chemistry of a biomimetic metal complex at the small rim (so-called "funnel complex") and the Huisgen cycloaddition. The intramolecular thermal reaction proceeds with a high efficiency, chemio- and regioselectivity, allowing the monofunctionalization of one aromatic unit among three. Thanks to the high yield and selectivity of the reaction, it can be applied successively twice on the same compound, which opens the route for inherently chiral calix[6]arenes. This methodology not only is of wide potential for obtaining and exploiting calix[6]arenes that are dissymmetrized at the large rim, but also stands as an exemplary strategy for the selective appending of a functional group on a host platform.     

Synthesis and Application of Low Molecular Weight PEI-Based Copolymers for siRNA Delivery with Smart Polymer Blends

Polyethylenimine (PEI) is a commonly used cationic polymer for small-interfering RNA (siRNA) delivery due to its high transfection efficiency at low commercial cost. However, high molecular weight PEI is cytotoxic and thus, its practical application is limited. In this study, different formulations of low molecular weight PEI (LMW-PEI) based copolymers polyethylenimine-g-polycaprolactone (PEI–PCL) (800 Da–40 kDa) and PEI–PCL–PEI (5–5–5 kDa) blended with or without polyethylene glycol-b-polycaprolactone (PEG–PCL) (5 kDa-4 kDa) are investigated to prepare nanoparticles via nanoprecipitation using a solvent displacement method with sizes ≈100 nm. PEG–PCL can stabilize the nanoparticles, improve their biocompatibility, and extend their circulation time in vivo. The nanoparticles composed of PEI–PCL–PEI and PEG–PCL show higher siRNA encapsulation efficiency than PEI–PCL/PEG–PCL based nanoparticles at low N/P ratios, higher cellular uptake, and a gene silencing efficiency of ≈40% as a result of the higher molecular weight PEI blocks. These results suggest that the PEI–PCL–PEI/PEG–PCL nanoparticle system could be a promising vehicle for siRNA delivery at minimal synthetic effort.     

Synthesis of libraries of thiazole, oxazole and imidazole-based cyclic peptides from azole-based amino acids. A new synthetic approach to bistratamides and didmolamides

Treatment of a 1 : 1 mixture of the thiazole-based amino acids 8a and 8b with FDPP-i-Pr(2)NEt in CH(3)CN gave a mixture of the cyclic trimers 14, 15, 16 and 17 and the cyclic tetramers 19 and 23 in the ratio 2 : 7 : 5 : 8 : 1 : 1 and in a combined yield of 70%. Separate coupling reactions between the bisimidazole amino acid 45 and the thiazole/oxazole amino acids 43a and 42a in the presence of FDPP-i-Pr(2)NEt led to the bisimidazole based cyclic trimers 55 and 57 respectively (54-57%) and to the cyclic tetramer 56 (8-11%). Similar coupling reactions involving the bisthiazole and bisoxazole amino acids 49 and 47 with the imidazole/oxazole/thiazole amino acids 41a, 42a and 43a gave rise to the library of oxazole, thiazole and imidazole-based cyclic peptides 58, 59, 60, 61, 62, 63, 64 and 65. A coupling reaction between the bisthiazole amino acid 49 and the oxazole amino acid 73 led to an efficient (36% overall) synthesis of bistratamide H (67) found in the ascidian Lissoclinum bistratum. Coupling reactions involving oxazolines with thiazole amino acids were less successful. Thus, a coupling reaction between the phenylalanine-based oxazoline amino acid 71a and either the thiazole amino acid 8a or the bisthiazole amino acid 74 gave only a 2% yield of the cyclic hexapeptide didmolamide A (4) found in the ascidian Didemnum molle. Didmolamide B (68) was obtained in 9% yield from a coupling reaction between 74 and the phenylalanine threonine amino acid 72, using either FDPP or DPPA.     

Drastic effects of the second coordination sphere on neutral vs. anionic guest binding to a biomimetic Cu(II) center embedded in a calix[6]aza-cryptand

A calix[6]arene capped by a tmpa unit provides Cu(ii) funnel complexes that, for the first time, accept not only neutral guest ligands but also anionic ones such as a hydroxide, which is due to a small conformational change in the calixarene cone.     

A hydrophilic cyclodextrin duplex forming supramolecular assemblies by physical cross-linking of a biopolymer

New beta-cyclodextrin (beta-CD) dimeric species have been synthesised in which the two CD moieties are connected by one or two hydrophilic oligo(ethylene oxide) spacers. Their complexation with sodium adamantylacetate (free adamantane) and adamantane-grafted chitosan (AD-chitosan) was then studied by different complementary techniques and compared with their hydrophobic counterparts that contain an octamethylene spacer. Isothermal titration calorimetry experiments have demonstrated that the use of hydrophilic spacers between the two CDs instead of aliphatic chains makes almost all of the CD cavities available for the inclusion of free adamantane. Investigation of the interaction of the CDs with AD-chitosan by viscosity measurements strongly suggests that the molecular conformation of the CD dimeric species plays a crucial role in their cross-linking with the biopolymer. The derivative doubly linked with hydrophilic arms, also called a duplex, has been shown to be a more efficient cross-linking agent than its singly bridged counterpart, referred to as a dimer. Hence, only 0.5 molar equivalents of the hydrophilic duplex with respect to adamantane was required to obtain the maximum viscosity, whereas in the case of the duplex with aliphatic spacers, the maximum viscosity was achieved with a [duplex]/[AD] ratio of about 1.7 (corresponding to a [CD]/[AD] ratio of 2.5), but with a higher value. To clarify the relationships between the molecular architecture and complexation properties, computational studies were also performed that clearly confirmed the importance of double bridging.