We consider whether or not protein chains in the HP model have unique or few optimal foldings. We solve the conjecture proposed by Aichholzer et al. that the open chain L2k−1 = (HP) k (P H) k−1 for k ≥ 3 has exactly two optimal foldings on the square lattice. We show that some closed and open chains have unique optimal foldings on the hexagonal and triangular lattices, respectively.
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