The helix, s Applequist, 1963) in which the Zimm-Bragg parameters u and s are defined respectively as the cooperativity factor for helix initiation, and the equi- librium constant for converting a coil residue to a helical ~~~~

Conformational parameters for amino acids in helical, beta-sheet, and random coil regions calculated from proteins.

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.

In solution, biopolymers commonly fold into well-defined three-dimensional structures, but only recently has analogous behavior been explored in synthetic chain molecules. An aromatic hydrocarbon backbone is described that spontaneously acquires a stable helical conformation having a large cavity. The chain does not form intramolecular hydrogen bonds, and solvophobic interactions drive the folding transition, which is sensitive to chain length, solvent quality, and temperature.

https://science.sciencemag.org/content/sci/277/5333/1793.full.pdf

Solvophobically Driven Folding of Nonbiological Oligomers

Methods for describing a discrete number of conformational states of amino acid residues in proteins are presented and used to investigate the topography of chain folding. The relative importance of short-range, medium-range and long-range interactions is discussed in the light of an analysis of the conformational states for the different amino acid residues in eight proteins of known structure. A prediction algorithm, which assigns four states to each residue of a protein chain (α-helix, extended structure, bend, or coil), has been developed from a consideration of both short- and medium-range interactions and applied to thirteen proteins of known three-dimensional structure. The prediction algorithm is simple to apply, and the assignment of α-helix and extended structure is considerably better than in most other predictive schemes. The prediction of chain reversal or bend regions was also better than with previous algorithms, but these assignments were not as good as those for α-helix and extended structure. The motivation for the development of this algorithm is not only to demonstrate the relative importance of short- and longer-range interactions but, more important, to begin to develop procedures for obtaining an approximate starting conformation for subsequent energy minimization to predict the three-dimensional structure of a protein. This procedure, as well as various other methods for the prediction of the backbone topography and conformational states of residues in proteins from the amino acid sequence, have been reviewed and evaluated by comparing the success of the methods to the success expected from a random assignment of conformational states.

Analysis of Conformations of Amino Acid Residues and Prediction of Backbone Topography in Proteins

Publisher Summary This chapter provides an overview of the quantum–mechanical work done in conformation of amino acid residues of proteins with the double objective of comparing the results with those of the previous empirical computations and with the constantly and rapidly increasing amount of experimental data. The most extensive applications have been in the field of proteins and their constituents, which exhibit the most significant conformational problems in biochemistry. The conformational properties of amino acid residues and polypeptides may be classified into three categories corresponding broadly to successive stages of development and refinement, and include—(1) empirical computations in the hard sphere approximation, (2) empirical computations in the partitioned potential energy approximation, and (3) quantum–mechanical computations. The best procedure, which allows the determination of the individual molecular orbitals, is the self-consistent field method. In the great majority of theoretical studies—whether empirical or quantum–mechanical—on the conformational properties of polypeptides, proteins, and their constituents the peptide bond has been considered as planar and trans; however, new developments in a number of directions have increased the interest of taking into consideration the possible nonplanarity of the peptide bonds.

Molecular orbital calculations on the conformation of amino acid residues of proteins.

Molecular theory of the helix-coil transition in polyamino acids, I. Formulation

A model for the helix-coil transition in specific-sequence copolymers of amino acids is presented. While the statistical weight matrix, required for the correlation of the states of four consecutive residues in this model, is 44 x 44, it is shown that this can be contracted to an 11 X 11 matrix, whose secular equation is the same as that for the Zimm-Bragg 8 x 8 matrix, which is contractable to a 4 X 4 matrix. The statistical weight of a residue in one of eight distinct states reflects the major intramolecular interactions responsible for helix stabilization. It is shown that, for copolymers, the form of the individual statistical weights is unique, rather than arbitrary as it is for homopolymers. While the 11 X 11 matrix is required for a copolymer, it reduces to a 4 X 4 matrix for a homopolymer. Numerical examples on a specific-sequence binary random copolymer indicate that the average helix contents, as determined by the 11 x 11 and by the Zimm-Bragg 2 x 2 matrix methods, differ in most cases by only a few per cent. Thus, the 2 X 2 matrix formulation can be used as a good approximation for the analysis of experimental data on polypeptide copolymers. However, the 11 X 11 matrix formulation has much more information than the 2 X 2 matrix formulation about the conformational state of each residue in a specific-sequence random copolymer. The new formulation has been applied to sperm whale myoglobin and hen egg white lysozyme to demonstrate how the more precise specification of the state of an amino acid residue in a denatured protein allows interesting correlations to be made with the corresponding three-dimensional structure in the native conformation. tatistical mechanical theories of the helix-coil transition S in homopolymers of amino a ~ i d s ~ ~ were originally formulated, explicitly or implicitly, in terms of specific models for the intramolecular interactions (e.g. , the assumption of the domination of hydrogen-bonding interactions for the stabilization of the right-handed a helix). The transition curves were computed in terms of two parameters (e .g . , u and s of Zimm and Bragg3), which were related to the intramolecular interactions. Among the various treatments, the nearest-neighbor interaction model (or the 2 X 2 matrix formulation) of Zimm and Bragg, being an approximate version of their more detailed model (requiring an 8 x 8 statistical weight matrix'^^), has the simplest mathematical structure, and has been applied successfully to the analysis of numerous experimental data since it yields numerical values of u and s which differ very little from those obtained with more elaborate theories involving larger matrices. Recently it was shown that, even though only short-range interactions were assumed to be important in the original theories, 3nevertheless the same mathematical formulation (e .g . , the 2 X 2 matrix treatment of Zimm and Bragg) still holds effectioely for homopolymers, even when longer range intramolecular interactions are involved, provided that the empirical parameters u and s are reinterpretedg so as to include contributions from these long-range interactions. However, as has been pointed out recently,10 the 2 X 2 matrix formulation does not apply to specific-sequence copolymers of amino acids; Le., because of the different chemical nature of the individual residues, it is not possible t o reinterpret u and s (which vary for each type of residue in a copolymer) so as to include the effect of the longer range interactions, as was done for the hom~polymer .~ Therefore, in this paper, we develop a proper form for the statistical weight matrix for a copolymer, based on the approximation that the statistical weight assigned to the ith residue is independent of the amino acid type of its neighbors,’O Le., side chain-backbone interactions (in a given solvent) determine the helix-forming power of each amino acid. Also, since the statistical weights in each matrix vary for each amino acid residue, it is not possible t o carry out a similarity tiansformation (as is done with homopolymers) to simplify the evaluation of the partition function for a specific-sequence copolymer; hence, the matrix multiplication has to be carried out explicitly. I t will be shown that we have t o use an 11 x 11 matrix, which can be reduced further to a 4 X 4 matrix only for a homopolymer, by using a similarity transformation which depends on the amino acid type of the homopolymer. Fortunately, as will also be shown, the numerical results obtained for both homopolymers and copolymers with the 11 X 11 matrix are close (within a few per cent) to those obtained with the Zimm-Bragg 2 x 2 matrix. Therefore, unless the Presently available Precimatrix to analyze experimental data for a copolymer. (1) This work was supported by research grants from the National Science Foundation (No. GB-28469X and GB-17388), from the National Institute of General Medical Sciences of the National Institutes of Health, U. S. Public Health Service (NO. GM-14312), from the Eli Lilly, Hoffmann-La Roche, and Smith Kline and French Grants Committees, and from Walter and George Todd. sion in experimental data is improved, one can use the 2 X 2 (2) (a) On leave of absence from the Deuartment of Physics, Faculty In section I, the model is described, and a mathematical . , , , of Science, University of Tokyo, Tokyo, Japan, 1967-1970-and summer, 1971 ; (b) National Research Council of Canada Postgraduate Fellow, 1971-1972. (3) B. H. Zimm and J. K. Bragg, J . Chem. Phys., 31, 526 (1959). (4) K. Nagai, J . Phys. Soc. Jup., 15, 407 (1960). (5) A. Miyake, J . Polym. Sci., 46, 169 (1960). (6) S. Lifson and A. Roig, J . Chem. Phys., 34, 1963 (1961). (7) The 8 X 8 matrix has been shown previously,8 in a heuristic fashion, to be contractable to a 4 X 4 matrix. A more systematic method for contracting statistical weight matrices is applied to this matrix in Appendix A. (8) D. Poland and H . A. Scheraga, “Theory of Helix-Coil Transitions in Biopolymers,” Academic Press, New York, N. Y., 1970, p 42. (9) D. Poland and H. A. Scheraga, Physiol. Chem. Phys., 1, 389 (1969). formulation for the partition function is presented in section 11. From this, the probability for each residue to be in one of eight possible states is calculated. In section 111, we present and discuss the numerical results based on this model and compare them with those obtained with the 2 X 2 matrix formulation of Zimm and Bragg. Finally, in section IV, we apply this model to two proteins, myoglobin and lysozyme, (IO) M. Go, N. Go, and H. A. Scheraga, J . Chem. Phys., 54, 4489 (1971). Vol. 4, No. 6, November-December 1971 SPECIFIC-SEQUENCE AMINO ACID COPOLYMERS 693

A Model for the Helix-Coil Transition in Specific-Sequence Copolymers of Amino Acids

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Molecular theory of the helix-coil transition in polyamino acids, I. Formulation

A Model for the Helix-Coil Transition in Specific-Sequence Copolymers of Amino Acids