Toward absolute density of states calculations for proteins

The density of states (DOS), which gives the number of conformations with a particular energy E, is a prerequisite in computing most thermodynamic quantities and in elucidating important biological processes such as the mechanism of protein folding. However, current methods for computing DOS of large systems such as proteins generally yield only the ratios of microstate counts for different energies, which could yield absolute conformation counts if the total number of conformations in phase space is known, thus motivating this work. Here, the total number of energy minima of 50-mer polyalanine, whose size corresponds to naturally occurring small proteins, was estimated under an all-atom potential energy function based on the cumulative distribution function (CDF) of conformational differences to be approximately 10(38). This estimate can place any DOS function, such as the Gaussian DOS distribution in the random energy model, on an absolute scale. Comparing the absolute conformational counts from a Gaussian DOS function with those from the CDF derived from quenched molecular dynamics ensembles shows that the former are far greater than the latter, indicating far fewer low-energy minima actually exist. In addition to showing how CDF and relative DOS calculations can yield absolute DOS for a discrete system, we also show how they can yield absolute DOS for continuous variable systems to a specified atomic variance. In the context of protein folding, knowing this phase-space "volume" of conformations in a DOS function, as well as characteristic transition times, constrains the set of possible folding mechanisms.