Unique optimal foldings of proteins on a triangular lattice.

A problem for unique protein folding was raised in 1998: are there proteins having unique optimal foldings for all lengths in the hydrophobic-hydrophilic (hydrophobic-polar; HP) model? To such a question, it was proved that on a square lattice there are (i) closed chains of monomers having unique optimal foldings for all even lengths and (ii) open monomer chains having unique optimal foldings for all lengths divisible by four. In this article, we aim to extend the previous work on a square lattice to the optimal foldings of proteins on a triangular lattice by examining the uniqueness property or stability of HP chain folding.We consider this protein folding problem on a triangular lattice using graph theory. For an HP chain with length n > 13, generally it is very time-consuming to enumerate all of its possible folding conformations. Hence, one can hardly know whether or not it has a unique optimal folding. A natural problem is to determine for what value of n there is an n-node HP chain that has a unique optimal folding on a triangular lattice.Using graph theory, this article proves that there are both closed and open chains having unique optimal foldings for all lengths >19 in a triangular lattice. This result is not only general from the theoretical viewpoint, but also can be expected to apply to areas of protein structure prediction and protein design because of their close relationship with the concept of energy state and designability.