Based on the dipole model of peptide groups developed in our earlier work [Liwo et al., Prot. Sci., 2, 1697 (1993)], a cumulant expansion of the average free energy of the system of freely rotating peptide‐group dipoles tethered to a fixed α‐carbon trace is derived. A graphical approach is presented to find all nonvanishing terms in the cumulants. In particular, analytical expressions for three‐ and four‐body (correlation) terms in the averaged interaction potential of united peptide groups are derived. These expressions are similar to the cooperative forces in hydrogen bonding introduced by Koliński and Skolnick [J. Chem. Phys., 97, 9412 (1992)]. The cooperativity arises here naturally from the higher order terms in the power‐series expansion (in the inverse of the temperature) for the average energy. Test calculations have shown that addition of the derived four‐body term to the statistical united‐residue potential of our earlier work [Liwo et al., J. Comput. Chem., 18, 849, 874 (1997)] greatly improves its performance in folding poly‐l‐alanine into an α‐helix. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 259–276, 1998

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A method is proposed to determine the conformational equilibrium of flexible polypeptides in solution, using the data provided by NMR spectroscopy and theoretical conformational calculations. The algorithm consists of the following three steps: (i) search of the conformational space in order to find conformations with reasonably low energy; (ii) simulation of the NOE spectrum and vicinal coupling constants for each of the low energy conformations; and (iii) determining the statistical weights of the conformations, by means of the maximum-entropy method, in order to obtain the best fit of the averaged NOE intensities and coupling constants to the experimental quantities. The method has been applied to two cyclic enkephalin analogs: DNS1-c-[D-A2bu2,Trp4,Leu5]enkephalin (ENKL) and DNS1-c-[D-A2bu2,Trp4,D-Leu5]enkephalin (ENKD). NMR measurements were carried out in deuterated dimethyl sulfoxide. Two techniques were used in conformational search: the electrostatically driven Monte Carlo method (EDMC), which results in extensive search of the conformational space, but gives only energy minima, and the molecular dynamics method (MD), which results in a more accurate, but also more confined search. In the case of EDMC calculations, conformational energy was evaluated using the ECEPP/3 force field augmented with the SRFOPT solvation-shell model, while in the case of MD the AMBER force field was used with explicit solvent molecules. Both searches and subsequent fitting of conformational weights to NMR data resulted in similar conformations of the cyclic part of the peptides studied. For both ENKL and ENKD a common feature of the low-energy solution conformations is the presence of a type II' or type IV beta-turn at residues 3 and 4; the ECEPP/3 force field also gives a remarkable content of type III beta-turn. These beta-turns are tighter in the case of ENKL, which is reflected in different distributions of the D-A2bu(N gamma H)...D-A2bu(CO) and D-A2bu(N gamma H)...Gly3(CO) hydrogen-bonding distances, indicating that the D-A2bu(N gamma H) amide proton is more shielded from the solvent than in the case of ENKD. This finding conforms with the results of temperature coefficient data of the D-A2bu(N gamma H) proton. It has also been found that direct (MD) or Boltzmann (EDMC) averages of the observables do not exactly conform with the measured values, even when explicit solvent molecules are included. This suggests that improving force-field parameters might be necessary in order to obtain reliable conformational ensembles in computer simulations, without the aid of experimental data.

A new method to surmount the multiple-minima problem in protein folding is proposed. Its underlying principle is to locate a group of large basins containing low-energy minima (hereafter referred to as superbasins) in the original energy surface. This is achieved by coupling the superbasins in the original surface to basins in a highly deformed energy surface (which contains a significantly reduced number of minima, compared to the original rugged energy surface). The distance scaling method (DSM) and the diffusion equation method (DEM) have been implemented to carry out the deformation. The procedure consists of macroiterations in which the parameter a, that controls the deformation, changes between two extreme values, a max and a min (a=0 corresponds to the original energy surface). The first macroiteration is initialized by imposing a maximum deformation on the original surface and then selecting 10 randomly generated conformations in the maximally deformed surface, whose energies are then minimized, usually leading to less than 10 minima; the next macroiterations are fed with the results of the previous ones. Each macroiteration consists of the following steps:  (i) reversal of the deformation from a max to a min; a limited search is carried out in the neighborhood of the minima at each stage of the reversal; (ii) collection of the new low-energy minima in the a min-deformed energy surface; (iii) back-tracking these minima up to a max while increasing the deformation. Steps i − iii are iterated until no new minima are found in the undeformed surface, or a predefined number of iterations is exceeded. In the initial macroiteration, a min is greater than 0, and a max is chosen so that the deformed energy surface has only a few minima. In each next macroiteration, the new a max is set at a min of the previous macroiteration, and a min is decreased, to reach 0 in the last macroiteration. The method was applied to united-residue polyalanine chains with a length of up to 100 amino acid residues, and to locate low-energy conformations of the 10-55 fragment of the B-domain of staphylococcal protein A.