An Efficient Deformation-Based Global Optimization Method for Off-Lattice Polymer Chains:  Self-Consistent Basin-to-Deformed-Basin Mapping (SCBDBM). Application to United-Residue Polypeptide Chains

Abstract

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.

Replica Exchange and Multicanonical Algorithms with the Coarse-Grained United-Residue (UNRES) Force Field

“…In Entropy Sampling, the probabilities of occurrence of a conformation x and a state with energy E, respectively, are defined as (10) (11) where n(E) and S(E) have similar meanings as described above. Equations 10 and 11 can be related to equations 8 and 9 by first setting β = 0 (i.e., temperature to infinity) in equations 8 and 9 and then multiplying the resulting probabilities by the weight factor exp[-S(E)/k B ].…”

Application of a chaperone‐based refolding method to two‐ and three‐dimensional off‐lattice protein models

“…This was done by a mixed procedure involving random sampling of the structure space, local search using the methods also incorporated in the Scheraga group's SCBDBM minimization procedure, 51 and short periods of molecular dynamics simulation followed by rapid quenching and conjugate gradient descent. When applied to the 6-mers and 7-mers of the two-dimensional model, these procedures found all the lowest-energy structures tabulated by Stillinger and Head-Gordon 42,43 and in fact found lower energy structures for certain sequences (the 6-mers AAAABA, AAABBA, ABABAB, BABABB, BABBAB, ABBBBB, and the 7-mers AAAAAAA, AAAB-BAB, ABABABA, ABBABBA, BAABBBB, ABBBBAB, BABBBBB).…”

“…The use here of a degree of local search on release from the chaperone is designed to help the system overcome these local energy barriers, but without undercutting the basic chaperone-derived principles being exploited. The methods used are a subset of those incorporated into the Scheraga group's SCBDBM global minimization procedure, 51 namely the addition of a small randomly chosen increment (here in the range [Ϫ10°, 10°]) to each internal angle, and the rigid rotation of part of the structure thorough a somewhat larger angle (here in the range [Ϫ30°, 30°]) about an axis passing through a randomly chosen pivot monomer. (The same techniques formed part of the search procedure for global minimum structures mentioned earlier, but there the angular search ranges were much larger.)…”

Conformation-Family Monte Carlo (CFMC): An Efficient Computational Method for Identifying the Low-Energy States of a Macromolecule

“…The first class is that of Monte Carlo methods with minimization, such as the original MCM method [1] [2] and the electrostatically driven Monte Carlo (EDMC) method [3]. The second class is that of deformation-based methods [4] such as the diffusion-equation method (DEM), the distance-scaling method (DSM), and their descendant as the self-consistent basin-to-deformed-basin method (SCBDBM) [5] [6]. The third class is that of genetic (recombination) algorithms such as conformational space annealing (CSA) [7] [8].…”

“…In the next section, we describe the CFMC method in detail. The following section describes the application of this method to the well-studied test protein, domain B of Staphylococcal protein A (SpA) [5] [8] [9]. CFMC performs much better than MCM and SCBDBM, and is roughly equal to CSA in efficiency.…”