Some Combinatorial Properties of S85

Protein folding prediction even in its simplest planar HP-model has been shown to be NPcomplete, but it is still a mathematical challenge to find the hidden secret(s) of the nature to resolve the famous Levinthal’s paradox i.e., “How can a protein choose in less than a millisecond an astronomical number of possible configurations?” [1],[2]. In fact several research works have been done to find lowest energy configurations for several large benchmark problems for two-dimensional hydrohobic-hydrophilic model. In this note we have studied property of the solutions of the benchmark sequence S85 which is {4H 4P 12H 6P 12H 3P 12H 3P 12H 3P 1H 2P 2H 2P 2H 2P 1H 1P 1H}, where total number of H’s and P’s are respectively 59 and 26. We have analyzed solutions of S85 that all have scores value 53 by using tabu search and move set by Lesh, Mitzenmacher and Whiteside in [3]. This value is conjectured to be -52 in [4],[5]. We have particularly investigated the role of the geometric shapes formed by H’s and the number of chambers in the folding of S85. Where a chamber is a sub-rectangular path (lattice path) that forms a corridor or one-end open labyrinth corridor in the folded protein strand. Figure 1 and 2 illustrate the role of these parameters in the folding. We hope that this may give another angle to study of the theoretical protein folding problem [6].