A theory of chemical reactions in biomolecules in solution: Generalized Langevin mode analysis (GLMA)

The generalized Langevin mode analysis (GLMA) is applied to chemical reactions in biomolecules in solution. The theory sees a chemical reaction in solution as a barrier-crossing process, similar to the Marcus theory. The barrier is defined as the crossing point of two free-energy surfaces that are attributed to the reactant and product of the reaction. It is assumed that both free-energy surfaces are quadratic or harmonic. The assumption is based on the Kim-Hirata theory of structural fluctuation of protein, which proves that the fluctuation around an equilibrium structure is quadratic with respect to the structure or atomic coordinates. The quadratic surface is a composite of many harmonic functions with different modes or frequencies. The height of the activation barrier will be dependent on the mode or frequency-the less the frequency, the lower the barrier. Hence, it is essential to decouple the fluctuational modes into a hierarchical order. GLMA is impeccable for this purpose. It is essential for a theoretical study of chemical reactions to choose a reaction coordinate along which the reaction proceeds. We suppose that the mode whose center of coordinate and/or the frequency changes most before and after the reaction is the one relevant to the chemical reaction and choose the coordinate as the reaction coordinate. The rate of reaction along the reaction coordinate is k_rate=ν⁡exp-ΔF^(†)/k_BT, which is similar to the Marcus expression for the electron transfer reaction. In the equation, ΔF^(†) is the activation barrier defined by ΔF^(†)≡F^(r)Q^†-F^(r)(Q_eq ^(r)), where F^(r)(Q_eq ^(r)) and F^(r)Q^† denote the free energies at equilibrium Q_eq ^(r) and the crossing point Q^†, respectively, both on the free energy surface of the reactant.