  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 which are
attributed to the reactant and product of the reaction. It is assumed that the
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, less the
frequency, lower the barrier. So, it is essential to decouple the fluctuational
mode into a hierarchical order. GLMA is impeccable for this purpose.
  It is essential for a theoretical study of chemical reactions to chose 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 , which is similar to the Marcus expression for the electron
transfer reaction. In the equation, is the activation barrier defined by ,
where and denote the free energies at equilibrium , and the crossing point ,
respectively, both on the free energy surface of the reactant.
