Efficient rare-event simulation for perpetuities

We consider perpetuities of the form

D=_B1exp(_Y1)+_B2exp(_Y1+_Y2)+?,$D=B_{1}exp\left(Y_1\right)+B_{2}exp(Y_1+Y_2)+?\text{,}$

where the _Yj$Y_j$’s and _Bj$B_j$’s might be i.i.d. or jointly driven by a suitable Markov chain. We assume that the _Yj$Y_j$’s satisfy the so-called Cramér condition with associated root _θ∗∈(0,∞)${\theta }_{\ast }\in (0,\infty )$ and that the tails of the _Bj$B_j$’s are appropriately behaved so that D$D$ is regularly varying with index _θ∗${\theta }_{\ast }$. We illustrate by means of an example that the natural state-independent importance sampling estimator obtained by exponentially tilting the _Yj$Y_j$’s according to _θ∗${\theta }_{\ast }$ fails to provide an efficient estimator (in the sense of appropriately controlling the relative mean squared error as the tail probability of interest gets smaller). Then, we construct estimators based on state-dependent importance sampling that are rigorously shown to be efficient.     

Synthesis of BINOL derived phosphorodithioic acids as new chiral Brønsted acids and an improved synthesis of 3,3′-disubstituted H8-BINOL derivatives

Original phosphorodithioic acid diesters were prepared according to an improved synthesis of 3,3′-disubstituted H₈-BINOL derivatives. In preliminary experiments, these new Brønsted acids were tested as organocatalysts in three reactions. They promoted the Nazarov cyclisation with mixed selectivities, the Mannich reaction with good enantioselectivity and they catalyzed efficiently the alkylation of N-acyliminium with enol silyl ether.     

3-trifluoromethanesulfonamido-pyrrolidine: a general organocatalyst for anti-selective mannich reactions

Mannich-type reactions of a glyoxylate imine with carbonyl compounds catalyzed by 3-trifluoromethanesulfonamidopyrrolidine proceed with high yields and anti-stereoselectivity. The catalyst is easily prepared and the transformation appears to be quite general accommodating aldehydes or ketones.     

3D structure determination of the Crh protein from highly ambiguous solid-state NMR restraints

In a wide variety of proteins, insolubility presents a challenge to structural biology, as X-ray crystallography and liquid-state NMR are unsuitable. Indeed, no general approach is available as of today for studying the three-dimensional structures of membrane proteins and protein fibrils. We here demonstrate, at the example of the microcrystalline model protein Crh, how high-resolution 3D structures can be derived from magic-angle spinning solid-state NMR distance restraints for fully labeled protein samples. First, we show that proton-mediated rare-spin correlation spectra, as well as carbon-13 spin diffusion experiments, provide enough short, medium, and long-range structural restraints to obtain high-resolution structures of this 2 x 10.4 kDa dimeric protein. Nevertheless, the large number of 13C/15N spins present in this protein, combined with solid-state NMR line widths of about 0.5-1 ppm, induces substantial ambiguities in resonance assignments, preventing 3D structure determination by using distance restraints uniquely assigned on the basis of their chemical shifts. In the second part, we thus demonstrate that an automated iterative assignment algorithm implemented in a dedicated solid-state NMR version of the program ARIA permits to resolve the majority of ambiguities and to calculate a de novo 3D structure from highly ambiguous solid-state NMR data, using a unique fully labeled protein sample. We present, using distance restraints obtained through the iterative assignment process, as well as dihedral angle restraints predicted from chemical shifts, the 3D structure of the fully labeled Crh dimer refined at a root-mean-square deviation of 1.33 A.