Starting from a kinetically foldable criterion for designing fast-folding structures, we have investigated the foldabilities of all possible sequences coded in two letters through an exhaustive enumeration of model chains of a 16-mer protein that we performed using a simple off-lattice model. From a set of 32,896 sequences, we found only 145 sequences that were foldable. Through it comparison of the geometrical similarities of those foldable Sequences, we reduced the corresponding 145 native structures to a structural set of 69 good candidates for target structures in the de novo design of fast-folding sequences. We make the following conclusions: (1) it preferred proportion of compositions exist for sequence design. (2) Foldable sequences having different numbers of hydrophobic residues possess very similar sequences. (3) The stability of some special structures toward mutations may be the origin of common protein structures; our results demonstrate that the presence of hydrophobic residues in certain positions of a sequence can result in firm and mutation-resistant skeletons. It appears that a simple, but robust, chain topology and structural symmetry lead to high designability. (c) 2005 Elsevier Ltd. All rights reserved.