Mean First-Passage Time Calculations for the Coil-to-Helix Transition: The Active Helix Ising Model†

The kinetics and thermodynamics of the coil-to-helix transition is studied using a one-dimensional “Zimm-Bragg ” Ising model. The mean first-passage time for the coil-to-helix transition is estimated within the “mean sequence ” approximation. A generalized mean first-passage time equation is derived where the transition rates may depend on the state of the system. The analytic expression for the mean first-passage time is evaluated, and the results are discussed as a function of energetic parameters, nucleation and propagation constants, peptide length, and the initial fraction of coil. The equilibrium thermodynamic properties of the model are shown to agree well with the Zimm-Bragg model, validating the mean sequence approximation. The time scales for helix formation are computed for a range of energetic parameters that determine the nucleation and propagation constants for the model. It is shown that, for a range of thermodynamically realistic parameters, the kinetic first-passage times are on the order of those measured experimentally. The mean first-passage time approach implicitly allows for the possibility of multiple helix nucleation sites and multiple helical domains and makes no assumptions regarding the unidirectionality of helix propagation. Comparison is made with the predictions of the “sequential kinetics ” model of Brooks and the “kinetic zipper ” model of Thompson et al. Extension of the model to the more general case of structure formation in proteins is discussed. 1