Ultrametricity in Protein Folding Dynamics.

The free energy of the transition state (TS) between two nodes of an ergodic Markov state model (MSM) can be obtained from the minimum cut, which is the set of edges that has the smallest sum of the flow capacities among all the possible cuts separating the two nodes. Here, we first show that the free energy of the TS is an ultrametric distance. The ultrametric property offers a way to simplify the MSM in a small number of states and, as a consequence, meaningful rate constants (free energy barriers) for the simplified MSM can be defined. We also present a new definition of the cut-based free energy profile (cbFEP), which is useful to check for the existence of a state for which the equilibration is much faster than the time to escape from it. From our analysis, a parallelism emerges between the minimum cut (maximum flow), and transition state theory (TST) or Kramers' theory.