A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient

We consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient, such as

$?F(x,u,Du,D^{2}u)=\lambda c(x)u+〈M(x)Du,Du〉+h(x)$

in a bounded domain with a Dirichlet boundary condition; here $\lambda \in \mathbb{R}$, $c,h\in L^p(\mathrm{\Omega })$, $p>n\ge 1$, $c?0$ and the matrix M satisfies $0<{\mu }_{1}I\le M\le {\mu }_{2}I$. Recently this problem was studied in the “coercive” case $\lambda c\le 0$, where uniqueness of solutions can be expected; and it was conjectured that the solution set is more complex for noncoercive equations. This conjecture was verified in 2015 by Arcoya, de Coster, Jeanjean and Tanaka for equations in divergence form, by exploiting the integral formulation of the problem. Here we show that similar phenomena occur for general, even fully nonlinear, equations in nondivergence form. We use different techniques based on the maximum principle.We develop a new method to obtain the crucial uniform a priori bounds, which permit to us to use degree theory. This method is based on basic regularity estimates such as half-Harnack inequalities, and on a Vázquez type strong maximum principle for our kind of equations.     

Panisuffrutin A,a highly degraded seco-triterpene derivative from Paeonia suffruticosa var. papaveracea (Andr.) Kerner

Panisuffrutin A (1), a highly degraded seco-triterpene derivative, together with the known palbinone, has been isolated from the whole plant Paeonia suffruticosa var. papaveracea (Andr.) Kerner. The structure with absolute configuration of 1 was determined via comprehensive NMR and MS analyses, as well as NMR and ECD calculations. A plausible biosynthetic pathway for 1 was proposed. Compound 1 showed weak cytotoxicity against Hela cancer cell line with an IC₅₀ value of 26.2 μM, while palbinone exhibited a moderate inhibition on NO production with an IC₅₀ of 18.3 μM.     

Hyphenone A,the first 3,3-diisoprenylated bicyclic polyprenylated acylphloroglucinols as Cav3.1 T-type calcium channel inhibitor from Hypericum henryi

Hyphenone A (1), a new type of bicyclic polyprenylated acylphloroglucinols (BPAPs) featured with an unprecedented 3,3-diisoprenylated 1,3,5-trione core, were characterized from the roots of Hypericum henryi together with two new congeners (hyphenones B–C, 2–3) and five known biosynthetic related PPAPs. Biogenetically, 1 should be derived from a novel 3,3,5,5-tetraisoprenylated MPAPs precursor. Their structures were elucidated by comprehensive spectroscopic data and X-ray diffraction. Moreover, 1–3 were identified as the first PPAPs type Ca_v3.1 T-type calcium channel (TTCC) inhibitors, with IC₅₀ values of 7.07, 6.19, and 5.47 μM, respectively.     

A practical access to new pyrrolizine carboxylates via KHMDS-catalyzed carbocyclizations

Easy and effective preparation of new 1H-pyrrolizine carboxylates was achieved with high efficiency via KHMDS-induced carbocyclization of N-alkynyl proline carboxylates under substantially mild conditions. Meanwhile, some trans-diiodoallylic N-proline carboxylates were obtained from N-propargyl proline carboxylates using molecular I₂ with or without KHMDS. This method is quite feasible in terms of practical and quick access to the pyrrolizines and their derivatives over the formation of carbanions.     

Dayaolingzhiols A−E,AchE inhibitory meroterpenoids from Ganoderma lucidum

Ganoderma mushrooms are widely used as effective medicines for treating and preventing chronic diseases. In this study, five new meroterpenoids, dayaolingzhiols A (1) and B (2) featuring a 6/6/6 ring system, dayaolingzhiols C?E (3–5) harboring a γ-lactone motif, along with eight known ones (6–13), were purified from G. lucidum. To clarify their chemical structures, spectroscopic and ECD and OR computational methods were used. In addition, the inhibition of all the new meroterpenoids against AChE was evaluated, disclosing that (+)-4 and (±)-5 possess potent AChE inhibitory activities with respective IC₅₀ values of 8.52?±?1.90?μM and 7.37?±?0.52?μM.     

Medicinal Potential of Isoflavonoids: Polyphenols That May Cure Diabetes

In recent years, there is emerging evidence that isoflavonoids, either dietary or obtained from traditional medicinal plants, could play an important role as a supplementary drug in the management of type 2 diabetes mellitus (T2DM) due to their reported pronounced biological effects in relation to multiple metabolic factors associated with diabetes. Hence, in this regard, we have comprehensively reviewed the potential biological effects of isoflavonoids, particularly biochanin A, genistein, daidzein, glycitein, and formononetin on metabolic disorders and long-term complications induced by T2DM in order to understand whether they can be future candidates as a safe antidiabetic agent. Based on in-depth in vitro and in vivo studies evaluations, isoflavonoids have been found to activate gene expression through the stimulation of peroxisome proliferator-activated receptors (PPARs) (α, γ), modulate carbohydrate metabolism, regulate hyperglycemia, induce dyslipidemia, lessen insulin resistance, and modify adipocyte differentiation and tissue metabolism. Moreover, these natural compounds have also been found to attenuate oxidative stress through the oxidative signaling process and inflammatory mechanism. Hence, isoflavonoids have been envisioned to be able to prevent and slow down the progression of long-term diabetes complications including cardiovascular disease, nephropathy, neuropathy, and retinopathy. Further thoroughgoing investigations in human clinical studies are strongly recommended to obtain the optimum and specific dose and regimen required for supplementation with isoflavonoids and derivatives in diabetic patients.     

Atomistic insights into the selective therapeutic activity of 6-(2,4-difluorophenoxy)-5-((ethylmethyl)pyridine-3-yl)-8-methylpyrrolo[1,2-a]pyrazin-1(2H)-one towards bromodomain-containing proteins

Cross-target effect has been one of the major mechanisms of drug toxicity, this has necessitated the design of inhibitors that are specifically tailored to target particular biomolecules. 6-(2,4-difluorophenoxy)-5-((ethylmethyl)pyridine-3-yl)-8-methylpyrrolo[1,2-a] pyrazin-1(2H)-one (Cpd38) is an inhibitor possessing high inhibition rate and tailored specificity towards bromodomain-containing protein 4 (BRD4). In this research, we used an array of computational techniques to provide insight at the atomistic level the specific targeting of BRD4 by Cpd38 relative to the binding of Cpd38 with E1A binding protein P300 (EP300); another bromodomain-containing protein (BCP). Comparatively, binding of Cpd38 improved the conformational stability and compactness of BRD4 protein when compared to the Cpd38 bound EP300. Also, Cpd38 induced a conformational change in the active site of BRD4 that facilitated a complementary pose between Cpd38 and BRD4 suitable for effective atomistic interactions. Expectedly, thermodynamic calculations revealed that the Cpd38-BRD4 system had higher binding energy (−36.11 Kcal/mol) than the Cpd38-EP300 system with a free binding energy of −15.86 Kcal/mol. Noteworthy is the opposing role Trp81 (acting as hydrogen bond acceptor) and Pro1074 (acting as hydrogen bond donor) found on the WPF and LPF loops respectively play in maintaining Cpd38 stability. Furthermore, the hydrogen bond acceptor/donator ratio was approximately 4:1 in Cpd38-BRD4 system compared with 2:1 in Cpd38-EP300 system. Taken together, atomistic insights and structural perspectives detailed in this report supplements the experimental report supporting the improved selectivity of Cpd38 for BRD4 ahead of other BCPs while providing leeway for the future design of BET selective agents with better pharmacological profile.