First Passage Times, Lifetimes, and Relaxation Times of Unfolded Proteins

The dynamics of proteins in the unfolded state can be quantified in computer simulations by calculating a spectrum of relaxation times which describes the time scales over which the population fluctuations decay to equilibrium. If the unfolded state space is discretized, we can evaluate the relaxation time of each state. We derive a simple relation that shows the mean first passage time to any state is equal to the relaxation time of that state divided by the equilibrium population. This explains why mean first passage times from state to state within the unfolded ensemble can be very long but the energy landscape can still be smooth (minimally frustrated). In fact, when the folding kinetics is two-state, all of the unfolded state relaxation times within the unfolded free energy basin are faster than the folding time. This result supports the well-established funnel energy landscape picture and resolves an apparent contradiction between this model and the recently proposed kinetic hub model of protein folding. We validate these concepts by analyzing a Markov state model of the kinetics in the unfolded state and folding of the miniprotein NTL9 (where NTL9 is the $N$-terminal domain of the ribosomal protein L9), constructed from a 2.9 ms simulation provided by D. E. Shaw Research.

Figure 1

The indicator function of the collective state $U\text{−}i$ is 1 when the trajectory is in state $U\text{−}i$ and zero otherwise. $t_1$, $t_2$, and $t_3$ are three realizations of the lifetimes of state $U\text{−}i$. Point $A$ is a typical starting point of the first passage time to state $i$.

Figure 2

Twenty-node MSM of NTL9 (the original and the rugged networks). (a) The $y$ coordinates of all the circles are the MFPTs calculated from Eq. (3). The $x$ coordinates of all the circles are calculated as shown in the legend. Both of the red and blue circles come from the original NTL9 network. The green circles are the results from the rugged NTL9 network, which are scaled down by fivefold to fit in the figure. The solid line is $y=x$. (b) Shows the implied time scales of the original network at lag time of 20 ns. (c) The implied time scale spectrum of the rugged network. The red lines in both (b) and (c) highlight the implied time scale corresponding to the dynamics between the $F$ and $U$ ensemble.