Total Synthesis of (+/-)-Deethylibophyllidine: Studies of a Fischer Indolization Route and a Successful Approach via a Pummerer Rearrangement/Thionium Ion-Mediated Indole Cyclization

The total synthesis of (+/-)-deethylibophyllidine is described, proceeding in eight steps from 4-(methoxyphenyl)ethylamine in 5% overall yield (Scheme 6). In terms of sequential annulation, the strategy involves the following operations: E --> DE --> ABDE --> ABCDE (Scheme 1). The key steps in the synthesis are the stereoselective formation of octahydroindol-6-ones by acid treatment of dihydroanisole derivatives, the regioselective Fischer indolization to obtain octahydropyrrolo[3,2-c]carbazoles, and the tandem process consisting of Pummerer rearrangement upon a beta-amino sulfoxide and thionium ion cyclization upon a beta-indole position of a 2,3-disubstituted indole to generate the quaternary spiro center. Attempts to effect the construction of the pentacyclic framework by means of Fischer indolization of the octahydropyrrolo[3,2,1-hi]indol-6-one resulted in failure (Scheme 2).     

Synthesis of microcin SF608

The first total synthesis of aquatic peptide microcin SF608 is described. Coupling of L-Hpla with the dipeptide L-Phe-L-Choi followed by coupling with agmatine and a deprotection step gave microcin SF608. In addition, the levorotatory character of L-Hpla (5) was thoroughly established, and the conformational analysis of L-Choi containing peptides 1 and 8-10 was performed using NMR spectroscopy to examine the cis-trans isomer equilibrium of the L-Phe-L-Choi amide bond.     

Nonuniform Dichotomies via Lyapunov Functions

For a nonautonomous linear equation x′ =  A(t)x we show how to characterize a nonuniform exponential dichotomy using strict Lyapunov functions. In particular, the stable and unstable subspaces are obtained from invariant families of cones determined by each Lyapunov function. We also obtain converse theorems, constructing explicitly a family of strict Lyapunov functions for each nonuniform exponential dichotomy. We emphasize that nonuniform exponential dichotomies include as a very particular case (uniform) exponential dichotomies.     

On the dynamics of the Szekeres system

The gravitational Szekeres differential system is completely integrable with two rational first integrals and an additional analytical first integral. We describe the dynamics of the Szekeres system when one of these two rational first integrals is negative, showing that all the orbits come from the infinity of ${\mathbb{R}}^4$ and go to infinity.     

Liouvillian and analytic integrability in a modified Philips model

We characterize the Liouvillian and analytic first integrals for a polynomial modified Phillips model.