Calculation of the entropy and free energy by the hypothetical scanning Monte Carlo method: application to peptides.

A new approach, the hypothetical scanning Monte Carlo (HSMC), for calculating the absolute entropy, S, and free energy, F, has been introduced recently and applied first to fluids (argon and water) and later to peptides. In this paper the method is further developed for peptide chains in vacuum. S is calculated from a given MC sample by reconstructing each sample conformation i step-by-step, i.e., calculating transition probabilities (TPs) for the dihedral and bond angles and fixing the related atoms at their positions. At step k of the process the chain's coordinates that have already been determined are kept fixed (the "frozen past") and TP(k) is obtained from a MC simulation of the "future" part of the chain whose TPs as yet have not been determined; when the process is completed the contribution of conformation i to the entropy is, S(i) approximately -ln Pi(k) TP(k). In a recent paper we studied polyglycine chains, modeled by the AMBER force field with constant bond lengths and bond angles (the rigid model). Decaglycine [(Gly)(10)] was studied in the helical, extended, and hairpin microstates, while (Gly)(16) was treated only in the first two microstates. In this paper the samples are increased and restudied, (Gly)(16) is also investigated in the hairpin microstate, and for (Gly)(10) approximations are tested where only part of the future is considered for calculating the TPs. We calculate upper and lower bounds for F and demonstrate that like for fluids, F can be obtained from multiple reconstructions of a single conformation. We also test a more realistic model of (Gly)(10) where the bond angles are allowed to move (the flexible model). Very accurate results for S and F are obtained which are compared to results obtained by the quasiharmonic approximation and the local states method. Thus, differences in entropy and free energy between the three microstates are obtained within errors of 0.1-0.3 kcal/mol. The HSMC method can be applied to a macromolecule with any degree of flexibility, ranging from local fluctuations to a random coil. The present results demonstrate that the difference in stability, DeltaF(mn)=F(m)-F(n) between significantly different microstates m and n, can be obtained from two simulations only without the need to resort to thermodynamic integration. Our long-term goal is to extend this method to any peptide and apply it to a peptide immersed in a box with explicit water.