The evaluation of the configurational partition function of a polypeptide molecule, with the internal rotation angles as variables, leads to an improved treatment of the phenomenon of helix-coil transition in polypeptide molecules. The conditional probabilities of occurrence of helical and coiled states of the peptide units are obtained in the form of a 3×3 matrix. The order of this matrix is the lowest possible for the model employed, and is derived by a logical procedure which serves to eliminate redundancies in the enumeration of states. The eigenvalues of this matrix yield the various molecular averages as functions of the degree of polymerization, temperature, and molecular constants. Explicit formulas are given for the degree of intramolecular hydrogen bonding, average number of helical sequences, and the distribution of their lengths, as well as the number average and the weight average of these lengths.
A new closed expression for the excluded volume effect of linear polymer molecules is developed with the aid of an equivalent ellipsoid model, in which the polymer chain with fixed ends is replaced by a uniform distribution of unconnected segments within an ellipsoid whose dimensions are chosen to give the correct principal radii of gyration of the chain. The linear expansion factor a for a chain of N segments is given bywhere a is the effective bond length and I'l is the binary cluster integral of a segment pair. In contrast to Flory's familiar equation, the ratio (a 6 -a 3 ) / N! is an increasing function of a instead of a constant. The new equation is in excellent agreement with the Monte Carlo calculations of Wall and Erpenbeck for a diamond lattice chain and also with existing viscosity data for various polymer solutions, provided that the expansion factor is evaluated from the equation ['1J/['1Jo=a 2 .43 in accord with the theories of Kurata-Yamakawa and Ptitsyn-Eisner. It is also shown that the present equation for a is in rather good agreement with the numerical solution of Fixman's differential equation.
SynopsisIn this paper two points are considered: the methods of evaluating the helical content 8 and the calculation of the parameters of the transition from experimental data and its interpretation. The parameter AH obtained is in good agreement with the calorimetric one and v is found to be independent of temperature and solvent and in agreement with the ordinarily accepted value for poly(y-benzyl-L-glutamate). The different methods of estimating 8 are discussed for both polypeptides.
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