  Under certain conditions, the dynamics of coarse-grained models of solvated
proteins can be described using a Markov state model, which tracks the
evolution of populations of configurations. The transition rates among states
that appear in the Markov model can be determined by computing the relative
entropy of states and their mean first passage times. In this paper, we present
an adaptive method to evaluate the configurational entropy and the mean first
passage times for linear chain models with discontinuous potentials. The
approach is based on event-driven dynamical sampling in a massively parallel
architecture. Using the fact that the transition rate matrix can be calculated
for any choice of interaction energies at any temperature, it is demonstrated
how each state's energy can be chosen such that the average time to transition
between any two states is minimized. The methods are used to analyze the
optimization of the folding process of two protein systems: the crambin
protein, and a model with frustration and misfolding. It is shown that the
folding pathways for both systems are comprised of two regimes: first, the
rapid establishment of local bonds, followed by the subsequent formation of
more distant contacts. The state energies that lead to the most rapid folding
encourage multiple pathways, and either penalize folding pathways through
kinetic traps by raising the energies of trapping states, or establish an
escape route from the trapping states by lowering free energy barriers to other
states that rapidly reach the native state.
