Folding protein models with a simple hydrophobic energy function:: The fundamental importance of monomer inside/outside segregation

The present study explores a "hydrophobic" energy function for folding simulations of the protein lattice model. The contribution of each monomer to conformational energy is the product of its "hydrophobicity" and the number of contacts it makes, i.e., E((h) over right arrow, (c) over right arrow) = -Sigma(i=1)(N) c(i)h(i) = -((h) over right arrow,(c) over right arrow) is the negative scalar product between two vectors in N-dimensional cartesian space: h = (h(1), ..., h(N)), which represents monomer hydrophobicities and is sequence-dependent; and (c) over right arrow = (c(1,) ..., c(N)), which represents the number of contacts made by each monomer and is conformation-dependent. A simple theoretical analysis shows that restrictions are imposed concomitantly on both sequences and native structures if the stability criterion for protein-like behavior is to be satisfied. Given a conformation with vector c; the best sequence is a vector h on the direction upon which the projection of (c) over right arrow - (c) over right arrow is maximal, where (c) over right arrow is the diagonal vector with components equal to (c) over right arrow the average number of contacts per monomer in the unfolded state. Best native conformations are suggested to be not maximally compact, as assumed in many studies, but the ones with largest variance of contacts among its monomers. i.e., with monomers tending to occupy completely buried or completely exposed positions. This inside/outside segregation is reflected on an apolar/polar distribution on the corresponding sequence. Monte Carlo simulations in two dimensions corroborate this general scheme. Sequences targeted to conformations with large contact variances folded cooperatively with thermodynamics of a two-state transition. Sequences targeted to maximally compact conformations, wh ich have lower contact variance, were either found to have degenerate ground state or to fold with much tower cooperativity.