We consider the question of how to design proteins. How can we find "good" amino acid sequences (i) that fold to a desired "target" structure as a native conformation of lowest accessible free energy and (ii) that will not simultaneously fold to many other conformations of the same free energy? Current protein designs often focus on helix propensities and turns. We focus here on designing the hydrophobicity. For a model of self-avoiding hydrophobic/polar chains on two-dimensional square lattices, geometric proofs and exhaustive enumerations show the following results. (i) The strategy hydrophobic residues inside/polar residues outside is not optimal. Placement of additional hydrophobic residues on the surface is often necessary. (ii) To avoid unwanted conformations, the designed sequence must have neither too many nor too few hydrophobic residues. (iii) The computational complexity of inverse folding appears to be in a different class than folding: unlike the folding problem, the design problem does not scale exponentially with chain length. Some design strategies, described here for the lattice model, produce good sequences and scale only linearly with chain length.
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