Kinetic study on unzippering of polynucleotides and order-order transition in polypeptides

Solutions are presented for N + 1 sequential and reversible first‐order reactions for which the magnitude of the reverse rate constant, kb, for all steps except the last is identical. Also the magnitude of the forward rate constant, kf, for all steps except the first and last is identical. The initial and final steps are nucleation reactions; therefore, the initial and final kf are modified by the factors σ′ and γ respectively. The final kb is modified by the factor γ σ. The ratio kb/kf is defined as s, which has the same meaning as s in the Zimm‐Bragg theory. The mathematical model is intended to apply to polymeric molecules of N segments and allows the calculation of the mole fraction of molecules in state i at any time t, Ci(t). A molecule in state i has i unreacted segments and N – i reacted ones. Because the reactions are sequential, all reacted segments are contiguous. Our numerical results show that when σ′ is much less than unity and the forward reaction is favored, the relaxation curve is sigmoidal. If, however, the forward and reverse reactions are equally favored (i.e., s ≃ 1) the relaxation curve is a straight line. When s and σ′ are near unity, the curve is exponential for a considerably large fraction of the reaction. Further, in the exponential for a considerably large fraction of the reaction. Further, in the exponential phase of the reaction, the relaxation time is proportional to N2 for highly cooperative systems (i.e., Nσ ≪ 1). As found by Pipkin and Gibbs, if N is sufficiently large and s is less than unity (e.g., N ≳ 50 and s ≃0.9) the relaxation curve is largely linear with a slope inversely proportional to N. Applications are given for the unwinding of double‐helical poly(A·U) and the order–order transition in poly‐L‐proline.