Publisher Summary In the proper environment, a polypeptide chain can fold spontaneously to the three-dimensional structure of a native protein, and despite the possible presence of barriers in conformational space, the chain can find its way around these barriers to reach the structure of lowest free energy. This chapter describes the spontaneous folding of proteins; in vitro complementation of protein fragments; flexibility of proteins in solution: effect of cross-links and ligands, synthetic analogs of proteins; experimental approaches to the study of conformation; and energetic factors determining protein folding. The interatomic interactions within the chain and between the chain and the solvent dictate the folding and the range of the forces involved, is such that near-neighbor interactions are dominant. Folding is envisaged as taking place by the formation of nucleation sites in various parts of the chain, in response to near-neighbor interactions; the various nucleation sites become stabilized, when they are brought into proximity, so that long-range interactions can become operative. However, the development of conformational energy calculation procedures will enable the three-dimensional structure of a native protein to be predicted from the knowledge of its amino acid sequence and its interactions with the solvent in which it is dissolved.

Experimental and theoretical aspects of protein folding.

We present a method to predict complex structural (conformational) transitions in irregular or disordered macromolecular systems, such as proteins or glasses, at the atomic level. Our method aims at rare events, which currently cannot be predicted with traditional molecular dynamics (MD) simulations, since these currently are limited to time scales shorter than a few nanoseconds. Given an initial conformation of the system, our method identifies one or more product states, which may be separated from the initial state by free energy barriers that are large on the scale of thermal energy. It also provides an approximate reaction path, which can be used to determine barrier heights or reaction rates with the usual techniques. The method employs an artificial potential that destabilizes the initial conformation and, thereby, lowers free energy barriers of structural transitions. As a result, transitions are accelerated and may be observed in MD simulations. An analytical estimate for the acceleration factor is given. The method is applied to two test systems, an argon microcluster and a simplified protein model. By these studies we demonstrated that our method is capable of shortening mean transition times from 0.5 \ensuremath{\mu}s (argon cluster) and 1.4 ns (protein model) to a few picoseconds. These results suggest that our method is particularly well suited to study biochemically relevant conformational motions in proteins at a microsecond time scale.

/pdf/predicting-slow-structural-transitions-in-macromolecular-4z8fik0eoc.pdf

Predicting slow structural transitions in macromolecular systems: Conformational flooding.

Revisiting free energy calculations: a theoretical connection to MM/PBSA and direct calculation of the association free energy.

(1980). Reverse Turns in Peptides and Protein. Critical Reviews in Biochemistry: Vol. 8, No. 4, pp. 315-399.

Reverse turns in peptides and proteins.

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 A, respectively. For 8-residue loops, 60 of 61 predictions have an rmsd of less than 3.0 A. The use of the colony energy is found to improve significantly the results obtained from the potential function alone. (The loop prediction program, “Loopy,” can be downloaded at http://trantor.bioc.columbia.edu.)

https://www.pnas.org/content/pnas/99/11/7432.full.pdf

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

Using an improved method for computing conformations of closed rings with symmetry, in conjunction with an improved empirical energy function, the conformational space of Gramicidin S is reexamined. The search for minimum energy conformations is confined to the subspace containing closed symmetric rings. A large number of initial conformations selected from that subspace is subjected to energy minimization or is eliminated in a sequence of steps designed to locate the global minimum-energy conformation. One conformation having distinctly low energy is found and is judged to be the global minimum-energy conformation. This conformation is of the beta-pleated sheet type and is in complete agreement with experimental data. Similar structures with beta-pleated sheet-type conformations have been proposed previously on the basis of less extensive examiniations of the conformational space; the condition of exact ring closure, and the extensive examination of conformational space, used here, establish this structure on a firm basis.

Use of a Symmetry Condition to Compute the Conformation of Gramicidin S

Calculation of the Conformation of cyclo-Hexaglycyl

The statistical weights of equilibrium conformations of macromolecules contain contributions from internal vibrations. An analysis of such vibrations, in the absence and presence of solvent, is presented from a quantum‐statistical‐mechanical point of view. Several classical approximations to the quantum‐mechanically correct expression, with different degrees of accuracy, are derived. In all of these approximations, all internal degrees of freedom of the polymer are divided into two classes: hard (bond stretching and bond angle bending) and soft (torsional rotations around single bonds, i.e., variation of dihedral angles). Since the hard variables oscillate many times before the soft variables change in value by an appreciable amount, the hard variables can be treated effectively as parameters (i.e., not as independent variables), which are (i) functions of the instantaneous values of the soft variables, or (ii) simply constants, the latter treatment being less accurate. In treatment (i) the molecule is re...

Analysis of the Contribution of Internal Vibrations to the Statistical Weights of Equilibrium Conformations of Macromolecules

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Use of a Symmetry Condition to Compute the Conformation of Gramicidin S

Calculation of the Conformation of cyclo-Hexaglycyl

Analysis of the Contribution of Internal Vibrations to the Statistical Weights of Equilibrium Conformations of Macromolecules