Theory of the Phase Transition between Helix and Random Coil in Polypeptide Chains

The transition between the helical and randomly coiled forms of a polypeptide chain is discussed by reference to a simple model that allows bonding only between each group and the third preceding. Two principal parameters are introduced, a statistical parameter that is essentially an equilibrium constant for the bonding of segments to a portion of the chain that is already in helical form, and a special correction factor for the initiation of a helix. A third parameter which specifies the minimum number of segments in a random section between two helical portions has only a minor effect on the results. The partition function for this model is handled in two alternative ways, either as a summation suitable for short chains, or in terms of the eigenvalues and eigenvectors of a characteristic matrix; the latter is more suitable for long chains. A transition from the random to the helical form is encountered as either the bonding parameter or the chain length is increased. The critical value of the bonding parameter is unity for long chains, while the sharpness of the transition depends on the initiation parameter. Depending on the values of the bonding parameter and the chain length, one of the following configurations dominates: random coils, single helices with occasional disorder at the ends, and for longer chains, helices occasionally broken by random sections. In rather narrow transition regions, mixtures of these forms may be found. A diagram is given that displays the relationships of these forms. The theory is compared with published data on polybenzyl-glutamate with fair agreement.