  Potential functions are critical for computational studies of protein
structure prediction, folding, and sequence design. A class of widely used
potentials for coarse grained models of proteins are contact potentials in the
form of weighted linear sum of pairwise contacts. However, these potentials
have been shown to be unsuitable choices because they cannot stabilize native
proteins against a large number of decoys generated by gapless threading. We
develop an alternative framework for designing protein potential. We describe
how finding optimal protein potential can be understood from two geometric
viewpoints, and we derive nonlinear potentials using mixture of Gaussian kernel
functions for folding and design. The optimization criterion for obtaining
parameters of the potential is to minimize bounds on the generalization error
of discriminating protein structures and decoys not used in training. In our
experiment we use a training set of 440 protein structures repre senting a
major portion of all known protein structures, and about 14 million structure
decoys and sequence decoys obtained by gapless threading. We succeeded in
obtaining nonlinear potential with perfect discrimination of the 440 native
structures and native sequences. For the more challenging task of sequence
design when decoys are obtained by gapless threading, we show that there is no
linear potential with perfect discrimination of all 440 native sequences.
Results on an independent test set of 194 proteins also showed that nonlinear
kernel potential performs well.
