  An effective potential function is critical for protein structure prediction
and folding simulation. Simplified protein models such as those requiring only
$C_\alpha$ or backbone atoms are attractive because they enable efficient
search of the conformational space. We show residue specific reduced discrete
state models can represent the backbone conformations of proteins with small
RMSD values. However, no potential functions exist that are designed for such
simplified protein models. In this study, we develop optimal potential
functions by combining contact interaction descriptors and local
sequence-structure descriptors. The form of the potential function is a
weighted linear sum of all descriptors, and the optimal weight coefficients are
obtained through optimization using both native and decoy structures. The
performance of the potential function in test of discriminating native protein
structures from decoys is evaluated using several benchmark decoy sets. Our
potential function requiring only backbone atoms or $C_\alpha$ atoms have
comparable or better performance than several residue-based potential functions
that require additional coordinates of side chain centers or coordinates of all
side chain atoms. By reducing the residue alphabets down to size 5 for local
structure-sequence relationship, the performance of the potential function can
be further improved. Our results also suggest that local sequence-structure
correlation may play important role in reducing the entropic cost of protein
folding.
