Asymptotics in the random assignment problem

Summary We show that, in the usual probabilistic model for the random assignment problem, the optimal cost tends to a limit constant in probability and in expectation. The method involves construction of an infinite limit structure, in terms of which the limit constant is defined. But we cannot improve on the known numerical bounds for the limit.Research supported by NSF Grant MCS90-01710     

The effect of solvent upon molecularly thin rotaxane film formation

We have investigated variations in molecularly thin rotaxane films deposited by solvent evaporation, using atomic force microscopy (AFM). Small changes in rotaxane structure result in significant differences in film morphology. The addition of exo-pyridyl moietes to the rotaxane macrocycle results in uniform domains having orientations corresponding to the underlying substrate lattice, while a larger, less symmetric molecule results in a greater lattice mismatch and smaller domain sizes. We have measured differences in film heights both as a function of the solvent of deposition and as a function of surface coverage of rotaxanes. Based on these observations we describe how the use of solvents with higher hydrogen-bond basicity results in films which are more likely to favour sub-molecular motion.     

Finite-time implications of relaxation times for stochastically monotone processes

Summary For a continuous-time finite state Markov process with stationary distribution , it is well-known thatP ⁱ (X ^t =j)- ^j isO(e ^-t ) ast, for a certain . For a stochastically monotone process for which the reversed process is also stochastically monotone, one can obtain bounds valid for allt. Precisely, exp(-t). The proof exploits duality for stochastically monotone processes.Research supported by National Science Foundation Grant MCS 84-03239     

Site-to-site peptide transport on a molecular platform using a small-molecule robotic arm

Peptides attached to a cysteine hydrazide ‘transporter module’ are transported selectively in either direction between two chemically similar sites on a molecular platform, enabled by the discovery of new operating methods for a molecular transporter that functions through ratcheting. Substrate repositioning is achieved using a small-molecule robotic arm controlled by a protonation-mediated rotary switch and attachment/release dynamic covalent chemistry. A polar solvent mixtures were found to favour Z to E isomerization of the doubly-protonated switch, transporting cargo in one direction (arbitrarily defined as ‘forward’) in up to 85% yield, while polar solvent mixtures were unexpectedly found to favour E to Z isomerization enabling transport in the reverse (‘backward’) direction in >98% yield. Transport of the substrates proceeded in a matter of hours (compared to 6 days even for simple cargoes with the original system) without the peptides at any time dissociating from the machine nor exchanging with others in the bulk. Under the new operating conditions, key intermediates of the switch are sufficiently stabilized within the macrocycle formed between switch, arm, substrate and platform that they can be identified and structurally characterized by ¹H NMR. The size of the peptide cargo has no significant effect on the rate or efficiency of transport in either direction. The new operating conditions allow detailed physical organic chemistry of the ratcheted transport mechanism to be uncovered, improve efficiency, and enable the transport of more complex cargoes than was previously possible.

Peptides are transported in either direction between chemically similar sites on a molecular platform, substrate repositioning is achieved using a cysteine hydrazide transporter module and a small-molecule robotic arm controlled by a rotary switch.     

A general strategy for the diastereoselective synthesis of 2,6-diaryl-3,7-dioxabicyclo[3.3.0]octane lignans

A strategy for the stereoselective synthesis of all the possible diastereoisomers of the 2,6-diaryl-3,7-dioxabicyclo[3.3.0]octane (furofuran) lignans from a single dihydrofuran precursor is described. The key steps involve a diastereocontrolled templated cationic cyclization followed by stereoselective reduction of the resulting methyl glycoside.     

Forward and backward pericyclic photochemical reactions have intermediates in common, yet cyclobutenes break the rules.

Photochemical pericyclic reactions are believed to proceed via a so-called pericyclic minimum on the lowest excited potential surface (S(1)), which is common to both the forward and backward reactions. Such a common intermediate has never been directly detected. The photointerconversion of 1,3-butadiene and cyclobutene is the prevailing prototype for such reactions, yet only diene ring closure proceeds with the stereospecificity that the Woodward-Hoffmann rules predict. This contrast seems to exclude a common intermediate. Using ultrafast spectroscopy, we show that the excited states of two cyclobutene/diene isomeric pairs are linked by not one, but by two common minima, p* and ct*. Starting from the diene side (cyclohepta-1,3-diene and cycloocta-1,3-diene), electrocyclic ring closure passes via the pericyclic minimum p*, whereas ct* is mainly responsible for cis-trans isomerization. Starting from the corresponding cyclobutenes (bicyclo[3.2.0]heptene-6 and bicyclo[4.2.0]octene-7), the forbidden isomer is formed from ct*. The path branches at the first (S(2)/S(1)) conical intersection towards p* and ct*. The fact that the energetically unfavorable ct* path can compete is ascribed to a dynamic effect: the momentum in C=C twist direction, acquired--such as in other olefins--in the Franck-Condon region of the cyclobutenes.