Evaluating conformational free energies: The colony energy and its application to the problem of loop prediction

Abstract

In this paper, we introduce a method to account for the shape of the potential energy curve in the evaluation of conformational free energies. The method is based on a procedure that generates a set of conformations, each with its own force-field energy, but adds a term to this energy that favors conformations that are close in structure (have a low rmsd) to other conformations. The sum of the force-field energy and rmsd-dependent term is defined here as the ''colony energy'' of a given conformation, because each conformation that is generated is viewed as representing a colony of points. The use of the colony energy tends to select conformations that are located in broad energy basins. The approach is applied to the ab initio prediction of the conformations of all of the loops in a dataset of 135 nonredundant proteins. By using an rmsd from a native criterion based on the superposition of loop stems, the average rmsd of 5-, 6-, 7-, and 8-residue long loops is 0.85, 0.92, 1.23, and 1.45 Å, respectively. force field ͉ energy minimization ͉ protein structure prediction P rotein loops are usually defined as segments of the polypeptide chain that do not contain regular units of secondary structure. Although some loops seem to serve as no more than connectors between secondary structure elements, others have been implicated as determinants of protein stability and folding pathways, while others may play important functional roles. The loop prediction problem involves finding the correct conformation for a given loop under the constraint that both ends are fixed through their connection to the rest of the protein. The problem has taken on considerable importance with the increased application of homology modeling methods in protein structure prediction. Although secondary structure elements can, in many cases, be predicted with considerable accuracy because they are often well conserved, sequence and structural variability are integral properties of many loops where, as in the case of antibodies, specificity differences among family members often reside. This variability makes the problem of loop prediction particularly complicated because, by its very nature, homology methods will often not be applicable. Indeed, loop prediction can to some extent be viewed as a mini ab initio folding problem, because the necessary information will not necessarily be found in databases. As such, the problem also serves as an important test of our understanding of the physical chemical principles that determine protein structure.Two general approaches have been applied to the prediction of loop conformation: database search and ab initio techniques. In the database search method (1-6), a library of segments derived from known protein structures is searched for conformations that fit the topological constraint of the loop stems. The stems correspond to the main-chain atoms that precede and follow the loop, but are not part of the loop itself. Loop candidates found in this way then can be evaluated by different criteria such as sequence...

Prediction of Protein Loop Conformations Using the AGBNP Implicit Solvent Model and Torsion Angle Sampling

“…For 9-residue loops, they report an average RMSD of 2.68 Å. 25 In comparison the average RMSD we have obtained with PLOP and AGBNP+ is 1.00 Å. De Bakker et al 62 generated loop conformations with their program RAPPER 63 and scored them with a knowledge-based potential and with a physics-based potential, AMBER/GBSA.…”

A hierarchical approach to all‐atom protein loop prediction

“…6,7,8 -17 . Among the various approaches to ab initio modeling are Monte Carlo Simulated Annealing, 18 molecular dynamics simulated annealing, 9,15 "random tweak," 7,8 and various dihedral angle sampling buildup methods, 6,10,14 sometimes supplemented with analytical loop closure methods. 17,19 -21 Our sampling algorithm is most closely related to the ab initio buildup procedures, although the detailed algorithm is entirely new.…”

“…At one extreme are methods in which energetics play little or no role, and scoring is accomplished by statistics-based functions (e.g., based on loop sequence 2 and stem matching, 2,5 or contact map representations 10 ). Fiser et al 9 used a mixture of molecular mechanics force-field terms and statistically derived terms for scoring, whereas Xiang et al 8 used a scoring function that included steric interactions, hydrophobicity, torsional energy, and hydrogen bonding, but ignored many electrostatic effects. One of the biggest challenges for a physics-based loop-scoring scheme is the treatment of solvent.…”

“…To facilitate comparison with previous work, we have chosen to study loop databases compiled previously by Fiser et al 9 and Xiang et al 8 . The loops were chosen in these previous studies from databases of high quality (2.0 Å resolution or better), diverse (maximum sequence identity 60% for Fiser et al and 20% for Xiang et al) crystal structures.…”

“…Xiang et al investigated all loops within their chosen protein data set; thus, the number of loops varies inversely with loop length due to natural abundance. Our results are reported as "global" RMSDs, obtained by superimposing the body of the protein; this choice is the same as that adopted by Xiang et al 8 and de Bakker et al 23 but differs from the "local" RMSD measure used by Fiser et al, 9 which provides no information about the orientation of the loop with respect to the body of the protein. If the objective is to use the structures for understanding interactions relevant to biological function and medicinal chemistry (e.g., docking candidate inhibitors), then accurate placement of the atoms of the loop in the context of the protein is of critical importance, and RMSD computed via loop stem superposition provides a reasonable measure of the effectiveness of the algorithm in accomplishing this task (ultimately, more direct measures, such as accuracy of docking into the resulting structure, would be even more relevant and insightful).…”