Combinatorial hierarchy

While in normal combinatorial peptide libraries (either chemical or phage display) each component has a unique sequence that is different from all others, in the cycloscan libraries all components have the same sequence, but differ in their conformation. This conformational diversity is generated in a dendrimeric hierarchy as shown exemplarily in Scheme 27 for the parent linear heptapeptide A-B-C-D-E-F-G. The diversity of the 1st order sublibrary (this nomenclature was adopted from Furka[468l) is based on the mode of cyclization. Excluding the head-to-tail cyclization there are seven different modes of cyclization that can be used for cycloscan three natural modes of cyclization and four modes of N-backbone cyclization. In addition there are five theoretical modes of C-backbone cyclization (see Scheme 1) which are not included in Scheme 27. 

We return to UCE prices in Section 5.1, in the context of an ascending-price combinatorial auction in which agents are interested in bundles of items. The primal-dual analysis is performed with respect to the hierarchy of extended LP formulations described in Section 2.3. 

Obviously, the preceding hierarchy of master density equations can also be closed at v = N. However, the product density equations may allow closure at a considerably lower value of r, which makes them much more attractive to solve than the master density equations. As pointed out earlier, even an analytical solution to the master density equation is not particularly valuable because of its combinatorial complexity. 

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