Comments on the one‐dimensional ising model and its application to the cis – trans equilibrium in poly‐ L ‐proline

A treatment of the Finite one‐dimensional Ising model is given in which four possible nearest‐neighbor interactions are taken into account, as well as the effect of the absence of neighbors at the ends of the chain. This represents the most general situation possible when interactions are limited to nearest neighbors and when the number of accessible states of a single residue is two. Some previous treatments of the Ising model are shown to be special cases of this problem. From the symmetry of the partition function it is found that the equilibrium transition curves for chains of all lengths intersect at their midpoints if and only if the free energy of both residue states is modified to the same extent when located at an end of the chain. The shift in transition midpoints with chain length generally observed for various one‐stranded systems is thus largely a measure of an end effect, and not a junction effect, as might seem to be the case from the theory of helix–coil equilibria. These considerations appear to be important in the cis—trans equilibrium in poly‐ L ‐proline, where the observations of Engel on short‐chain samples suggest that the effects at junctions and at chain ends must differ significantly. It is concluded from his observations that a cis unit at the end of a chain has a less favorable free energy than a trans unit at the end. This is attributed to the fact that there are no van der Waals contacts between three residues at the end of the relatively compact cis helix and the missing adjacent turn of the helix, while the residues in the relatively open trans helix are little affected by their location at the end of the chain.