Eigenvalue spectrum of the master equation for hierarchical dynamics of complex systems

We explored the eigenvalue spectra of the kinetic matrix which defines the master equation for the complex kinetics of the analogous polypeptides (linear Ala6, cyclic Ala6, and charged Ala6). For each system we obtained the entire eigenvalue spectrum as well as the histograms of the weighted eigenvalue spectra, where each relaxation mode is weighted by the overlap between the initial probability vector and the corresponding eigenvector. It was found that the spectra of the weighted eigenvalues were significantly filtered in comparison to those of the unweighted eigenvalues, indicating that the decay is described by a small number of eigenvalues. The important eigenvalues which are extracted from the weighted eigenvalues spectra are in good agreement with the characteristic lifetimes for the kinetics of each system, as found by the fitting of the energy relaxation temporal profiles to multiexponential functions. Moreover, a partial correlation is found between the relative heights of the contributions of the important eigenvalues and the preexponential factors obtained by the fitting. In addition, we applied the spectra of the weighted eigenvalues to study the effect of the initial population distribution on the dynamics and also to infer which minima provide the dominant contributions to a specific relaxation mode. From the latter results one can infer whether the multiexponential relaxations represent sequential or parallel processes. This analysis establishes the interrelationship between the topography and topology of the energy landscapes and the hierarchy of the relaxation channels.