Describing protein folding kinetics by molecular dynamics simulations.: 2.: Example applications to alanine dipeptide and β-hairpin peptide

In this work we demonstrate the use of a rigorous formalism for the extraction of state-to-state transition functions as a way to study the kinetics of protein folding in the context of a Markov chain. The approach is illustrated by its application to two different systems: a blocked alanine dipeptide in a vacuum and the C-terminal beta-hairpin motif from protein G in water. The first system displays some of the desired features of the approach, whereas the second illustrates some of the challenges that must be overcome to apply the method to more complex biomolecular systems. For both example systems, Boltzmann weighted conformations produced by a replica exchange Monte Carlo procedure were used as starting states for kinetic trajectories. The alanine dipeptide displays Markovian behavior in a state space defined with respect to phi-psi torsion angles. In contrast, Markovian behavior was not observed for the P-hairpin in a state space where all possible native hydrogen bonding patterns were resolved. This may be due to our choice of state definitions or sampling limitations. Furthermore, the use of different criteria for hydrogen bonding results in the apparent observation of different mechanisms from the same underlying data: one set of criteria indicate a zipping type of process, but another indicates more of a collapse followed by almost simultaneous formation of a large number of contacts. Analysis of long-lived states observed during the simulations of the beta-hairpin suggests that important aspects of the folding process that are not captured by order parameters in common use include the formation of non-native hydrogen bonds and the degree and nature of salt bridge formation.