Chain equivalence

In the course of experiments directed at the synthesis of open-chained equivalents of macrocyclic polyethers (see Chap. 7), Rasshofer and Vogtle found that 9 or its derivatives were formed rather than the desired compounds . Although the yields obtained by this method are modest, the facts that the reaction is conducted without the use of high dilution and in one-pot recommend it. 

A second source of inspiration for studying the open-chained equivalents of crown ethers was the observation that a number of naturally occurring antibiotics enhance cation transport and bear a structural similarity to open-chained crown ethers. A number of groups have examined neutral synthetic ionophores and a variety of novel cation carriers is now available. This is discussed in Sect. 7.4, below. 

The Prelog-Djerassi lactone (abbreviated here as P-D lactone) was originally isolated as a degradation product during structural investigations of antibiotics. Its open-chain equivalent 3 is typical of the methyl-branched carbon chains that occur frequently in macrolide and polyether antibiotics. The compound serves as a test case for the development of methods of control of stereochemistry in such polymethylated structures. There have been more than 20 different syntheses of P-D lactone.24 We focus here on some of those that provide enantiomerically pure product, as they illustrate several of the methods for enantioselective synthesis.25... 

Although acyclic polyethers, e.g., polyethylene glycols, form less stable solvates than the cyclic counterparts, they are also able to act as catalysts in biphasic systems. Typical structures of open-chain equivalents of crown ethers and cryptates are glymes32-34 (3) polyethylenamines35,36 (4) poly-podes37,38 (5) lariat ethers39,40 (6) and octopus41 (7). 

Vogtle et al.28 292 293 have also described open-chain equivalents of crown ethers (193) and cryptates (194) having heterocyclic substituents, such as quinoline. 

The intense interest in y - am i n o - (i -hydroxy acid related peptides has led to a great deal of effort devoted to their synthesis. Statine analogues Ahppa (11), Achpa (12), and Dahoa (13) (Scheme 2), which correspond to the replacement of the chain equivalent of leucine in statine by those of phenylalanine, cyclohexylalanine, and lysine, respectively, were used to synthesize inhibitors of pepsin, renin, and penicillinopepsinJ1314] With the objective of developing highly specific inhibitors, C2-substituted analogues have also been introduced into short, substrate-derived inhibitors, e.g. the a,a-difluoro statine 14 in renin inhibitors,1151 and N -substituted statine derivatives like 15 in HIV-1 protease inhibitors.[16]... 

Consider first the two chain problem and for simplicity let us make both chains equivalent. Then four coupled equations (1) can be reduced to three by substituting the variables U = g1Li/2vv, W = g /2-nv, V = (91ii-292ii+2g2ij leading to... 

The shorter-chain reversed-phase packings bonded with or Cg ligands were initially used for peptide analysis, but now the standard octadecyl packings are most widely used. From the standpoint of the separation, any chain length can be used successfully, albeit with some differences in selectivity. The longer-chain packings are hydrolytically more stable than their shorter-chain equivalents. 

The branching coefficient as defined by Flory is a = the probability that a certain branched unit will be joined to a second branched unit rather than to a terminal group. For example, for a trifunctional monomer if a = 1/2 the molecule is a continuous chain equivalent in theory to a gel. In this case a = 1/2 is the critical condition defining the start of the formation of an infinite tridimensional network. 

A finite state space model is irreducible if every state is reachable from every other state in a finite time , where time in this context refers to the number of iterations of the chain. Equivalently, we say the model is irreducible if all states... 

With this we have for quantization the chain equivalences ... 

A chain complex C is chain contractible if it is chain equivalent to Oj a chain homotopy r lc 0 C - C is a chain contraction of C... 

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