A Symmetry-Free Subspace for Ab initioProtein Folding Simulations

Ab initio protein structure prediction usually tries to find a ground state in an extremely large phase space. Stochastic search algorithms are often employed by using a predefined energy function. However, for each valid conformation in the search phase space, there are usually several counterparts that are reflective, rotated or reflectively rotated forms of the current conformation, imprecisely called isometric conformations here. In protein folding, these isometric conformations correspond to the different rotation states caused by admissible protein structure transitions. In structure prediction, these isometric conformations, owning the same energy value, not only significantly increase the search complexity but also degrade the stability of some local search algorithms. In this paper, we will prove that there exists a subspace that is unique (no two conformations in the space are isometric) and complete (for any valid conformation, there exists a corresponding conformation in the subspace that is a reflective or rotated form of it). We demonstrate that this subspace, which is about 1/24 of the conventional search space in the 3D lattice model and 1/8 in the 2D model contains the lowest energy conformation, and all other isometric lowest energy forms can then be obtained by protein rotation. Our experiments show that the subspace can significantly speed up existing local search algorithms.