Protein folded states are kinetic hubs

Abstract

Understanding molecular kinetics, and particularly protein folding, is a classic grand challenge in molecular biophysics. Network models, such as Markov state models (MSMs), are one potential solution to this problem. MSMs have recently yielded quantitative agreement with experimentally derived structures and folding rates for specific systems, leaving them positioned to potentially provide a deeper understanding of molecular kinetics that can lead to experimentally testable hypotheses. Here we use existing MSMs for the villin headpiece and NTL9, which were constructed from atomistic simulations, to accomplish this goal. In addition, we provide simpler, humanly comprehensible networks that capture the essence of molecular kinetics and reproduce qualitative phenomena like the apparent two-state folding often seen in experiments. Together, these models show that protein dynamics are dominated by stochastic jumps between numerous metastable states and that proteins have heterogeneous unfolded states (many unfolded basins that interconvert more rapidly with the native state than with one another) yet often still appear two-state. Most importantly, we find that protein native states are hubs that can be reached quickly from any other state. However, metastability and a web of nonnative states slow the average folding rate. Experimental tests for these findings and their implications for other fields, like protein design, are also discussed.

Fig. 1.

Three representative networks each having unfolded state(s) (U and U_i), intermediates (I_i), and a native state (N). S has a single pathway, P has parallel pathways, and H has a heterogeneous unfolded state.

Fig. 2.

Distributions of the first folding times for the simple networks S, P, and H are shown in (A), (B), and (C), respectively. The blue lines are exponential fits to the data after the initial lag phase.

Fig. 3.

Relaxation of villin from 500 state model. Distributions of the MFPTs from (A) unfolded states to the native state and (B) between unfolded states. (C) Relaxation kinetics with a 10∶1 signal-noise ratio (black curve with Gaussian noise) and a single exponential fit (blue curve with τ = 810 ns).

Fig. 4.

Schematic diagrams of funnel and native hub models having unfolded states (U), intermediates (I), and native states (N). (A) A network description of a folding funnel with nodes corresponding to individual conformations and a bottleneck near the native state. (B) A native hub model with metastable nodes. The size of each node in (B) is correlated with its equilibrium probability and the connectivity falls off as one moves away from the native state.

Fig. 5.

Distance between the final villin MSM and MSMs constructed from subsets of the data (varying trajectory length and number of trajectories). Distance is measured by a relative entropy metric (SI Text). Black lines are contours of equal amounts of data. No data was available for the upper-right portion of the graph.