How to determine the three dimensional structures of the protein coded by a primary peptide sequence is considered one of the most important problems in biophysics. Suggested solutions to this problem generally involve a simplified structural model for proteins, a Hamiltonian, and a dynamic algorithm with which to search for the lowest energy state of the model. Perhaps the simplest Hamiltonian is the so-called random energy model (REM). While the REM can provide some insights it does not permit one to examine specific native folding as seen in real proteins. For this reason we have chosen to study the more complicated model in which the Hamiltonian is an energy matrix with specific interactions between residues and we allow contacts to have the dependencies that naturally arise because some will involve the same residue. Our studies have thus far led to the following developments: i) The energy spectrum is shown to be Gaussian in the limit of realistic numbers of contracts as seen in proteins. ii) We have found a general statistical method of computing the probability of a unique native state when the energy distribution from which the energies of the compact conformations are to be sampled is known. iii) We have developed a method based on random walks with barriers which allow us to give an accurate picture of the extreme tails of the energy density for model molecules with energies determined by an energy matrix. We have used the method to study the amount of information in different structural features of proteins such as secondary structure, tertiary contacts, and solvent accessibility. In another study we are supplying our statistical prediction methods to estimate the probability that a particular energy generated by the random walk will correspond to a native (dominant Boltzman probability) state. The sequence and structure pairs that correspond to native states are then used to test the recovery of the energy matrix from the contact data. We expect in this way to be able to examine the validity of such statistical recover methods.
