Relaxation in biological macromolecules: Properties of some exact solutions

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Abstract
We develop a modification in the Glauber theory for relaxation in the one dimensional Ising model in order to treat finite chains with two boundaries (ends). We also present an analysis of an exact solution recently obtained by Schwarz and Poland for the reversible unraveling from the ends of a homogeneous multistrand complex with an irreversible final step (chain dissociation). By modification of the Galuber theory, we develop an exact solution for relaxation in finite chains of homopolyamino acids for a specific choice of transition probabilities. For N the chain length and s the Zimm–Bragg helix stability constant, the solution describes the relaxation for arbitrary N from an arbitrary initial value of s (temperature) to a final state at s=1. It is shown that the relaxation process can be factored exactly into a diffusion and an exponential decay term which dominate respectively in the limit of short and long chain lengths, while Glauber’s solution with periodic boundary conditions decays by the exponential term only. We find that, to a good approximation, a universal set of relaxation curves exist for appropriate scaled variables. In contrast to the behavior of polyamino acids, the unwinding of multistrand complexes can be viewed as involving transitions between free energy states with increasing degeneracy as the free energy (paralleling the degree of unwinding) increases. The rate of dissociation is found to be dominated by the rate of passage from the low probability highest free energy level, to the completely dissociated molecule.