The statistics of unique native states for random peptides

Given a probability distribution from which the energy spectrum of a random peptide is to be sampled, we derive a general expression for the probability that such a peptide will fold to a unique native state and for the probability distribution of the native energy. This latter result allows us to localize the energy of folding based on model parameters and is one advantage of our formulation. Evidence from both the lattice theory of proteins and protein threading experiments suggest that the energy spectrum for the compact states of a peptide chain is Gaussian in form. For this reason we have derived from the more general framework the specific formulas that apply in the Gaussian case, where one requires only the number of states and the variance of the Gaussian distribution in order to apply the theory. This simplicity allows us to perform calculations that we compare with calculations previously made by others based on statistical thermodynamics. We find qualitative agreement, but a significant correction to prior estimates of folding probability derived from the Gaussian assumption is necessary.