From laws of inference to protein folding dynamics

Protein folding dynamics is one of major issues constantly investigated in the study of protein functions. The molecular dynamic (MD) simulation with the replica exchange method (REM) is a common theoretical approach considered. Yet a trade-off in applying the REM is that the dynamics toward the native configuration in the simulations seems lost. In this work, we show that given REM-MD simulation results, protein folding dynamics can be directly derived from laws of inference. The applicability of the resulting approach, the entropic folding dynamics, is illustrated by investigating a well-studied Trp-cage peptide. Our results are qualitatively comparable with those from other studies. The current studies suggest that the incorporation of laws of inference and physics brings in a comprehensive perspective on exploring the protein folding dynamics.

Figure 1

(Color) The NMR structure of Trp-cage (PDB code is 1L2Y). The structure shows ${\text{Trp}}^6$ is caged inside the hydrophobic pocket formed by three proline.

Figure 2

An example of applying EMD to decompose the potential energy profile of a 40 ns long Trp-cage MD simulation. The first row is the raw potential energy profile. The last row “res” is the residual component after the decomposition.

Figure 3

(Color) The ranking scheme of seventy thousand Trp-cage structures obtained from OPTICS. The definition of reachability distance is given in the text. To illustrate the differences of Trp-cage structures at different ranks, three cartoon representations generated from Swiss-DeepView [28] are shown.

Figure 4

The potential energy profile of seventy thousand Trp-cage structures, which is ranked according to the ranking scheme obtained from OPTICS (the gray line). The dark line represents the smoothed potential energy obtained from the EMD-based method.

Figure 5

(Color) Seventy thousand Trp-cage structures are plotted in reachability distance vs smoothed potential energy diagram. Seven cartoon representations of Trp-cage structures generated from Swiss-DeepView [28] at different locations in the diagram are presented to show the structural differences at various locations of the diagram.

Figure 6

(Color) The evolution of ME PDF, Eq. (6), given an initial state $B^1=2.2\text{Å}$ and $B^2=-312\text{kcal}/\text{mole}$. The plot only shows the results at the initial, 20th, 40th, and the final step along with the corresponding averaged Trp-cage cartoon representations generated by VMD [16]. The figure shows that the probabilities at lower ranks are gradually increased while at higher ranks are gradually decreased along the trajectory.

Figure 7

(Color) Entropic folding trajectories of Trp-cage given three various initial starting structures. The three trajectories are plotted on the three-dimensional entropy surface. In the bottom, the same trajectories are projected on the two dimensional entropy contour map. The averaged initial (bottom) and final (top) structures generated by VMD [16] are also presented for comparisons.

Figure 8

The RMSD of averaged structures along the three trajectories and the NMR structure of Trp-cage. The lines with hollow square, circle and triangle are from trajectories (a), (b), and (c), respectively. Note that we only label the data points by symbols at every 4 steps.