Origin of subdiffusions in proteins: Insight from peptide systems

Subdiffusive kinetics are popular in proteins and peptides as observed in experiments and simulations. For protein systems with diverse interactions, are there multiple mechanisms to produce the common subdiffusion behavior? To approach this problem, long trajectories of two model peptides are simulated to study the mechanism of subdiffusion and the relations with their interactions. The free-energy profiles and the subdiffusive kinetics are observed for these two peptides. A hierarchical plateau analysis is employed to extract the features of the landscape from the mean square of displacement. The mechanism of subdiffusions can be postulated by comparing the exponents by simulations with those based on various models. The results indicate that the mechanisms of these two peptides are different and are related to the characteristics of their energy landscapes. The subdiffusion of the flexible peptide is mainly caused by depth distribution of traps on the energy landscape, while the subdiffusion of the helical peptide is attributed to the fractal topology of local minima on the landscape. The emergence of these different mechanisms reflects different kinetic scenarios in peptide systems though the peptides behave in a similar way of diffusion. To confirm these ideas, the transition networks between various conformations of these peptides are generated. Based on the network description, the controlled kinetics based only on the topology of the networks are calculated and compared with the results based on simulations. For the flexible peptide, the feature of controlled diffusion is distinct from that of simulation, and for the helical peptide, two kinds of kinetics have a similar exponent of subdiffusion. These results further exemplify the importance of the landscape topology in the kinetics of structural proteins and the effect of depth distribution of traps for the subdiffusion of disordered peptides.

Figure 1

Illustrations of (a) random network and (b) fractal network. For proteins, the vertices represent the conformations and the edges indicate the jumps between different conformations.

Figure 2

The snapshots of the peptides ${\mathrm{GS}}_4$ and ${\mathrm{A}}_8$. The carbon, oxygen, and nitrogen atoms are represented as grey, red, and blue spheres, respectively. The $N$ and $C$ termini of these two peptides are capped with acetyl and $N$-methylamide, respectively.

Figure 3

Ramachandran map based on the backbone dihedral angle $\varphi$ and $\psi$ for (a) glycine, (b) serine, and (c) alanine. The unit of energy is $k_{B}T$. In each plot, several regions are marked with numbers. Each region represents a state of residue with similar structural features. For the residue glycines, four states are introduced, and for serines and alanines, three states are defined.

Figure 4

The figure gives the evolutions of the end-to-end distance $Q_{\text{ee}}$ for (a) ${\mathrm{GS}}_4$ and (b) ${\mathrm{A}}_8$, the evolution of the number $N_{\text{HB}}$ of intrapeptide hydrogen bonds for (c) ${\mathrm{GS}}_4$ and (d) ${\mathrm{A}}_8$, the histogram of the number $N_{\text{HB}}$ during the simulation for (e) ${\mathrm{GS}}_4$ and (f) ${\mathrm{A}}_8$ and the free-energy profiles of the peptides (g) ${\mathrm{GS}}_4$ and (h) ${\mathrm{A}}_8$ with the end-to-end distance as the reaction coordinate. Some typical snapshots of the peptides at certain times are shown (for ${\mathrm{GS}}_4$, 100 ns of trajectories are used for clarity of presentation).

Figure 5

This figure gives the TA-MSD (upper row) and autocorrelation functions (lower row) for peptides ${\mathrm{GS}}_4$ (left column) and ${\mathrm{A}}_8$ (right column). Green lines in (c) and (d) are the fittings for 1000 ns trajectories with Eq. (11).

Figure 6

This figure gives the spectra of eigenvalues of covariance matrix for the peptides (a) ${\mathrm{GS}}_4$ and (b) ${\mathrm{A}}_8$, (c) the root mean square of fluctuations of residues in these two peptides (black for ${\mathrm{GS}}_4$ and red for ${\mathrm{A}}_8$), and the evolution in the subspaces spanned with essential eigencomponents for (d) ${\mathrm{GS}}_4$ and (e) ${\mathrm{A}}_8$ (for ${\mathrm{GS}}_4$, only 100ns of trajectories are used for clarity of presentation). The largest gaps of eigenvalues for these peptides are marked with red arrows in (a) and (b).

Figure 7

This figure gives the mean square of displacement as a function of lag time in the subspace, (a) the coordinate of end-to-end distance (EED), and (b) spanned with essential components based on xPCA (the data are also shown in the insets with normal scale), and the exponents ${\alpha }_{\text{local}}$ related to local diffusion motions for the peptides (c) ${\mathrm{GS}}_4$ and (d) ${\mathrm{A}}_8$ (the onset time ${\widehat{\tau }}^c$ and the escaping time ${\tau }^e$ for a certain valley are marked accordingly).

Figure 8

This figure gives the mean square of displacement as a function of lag time in the subspace spanned with essential components based on dPCA.

Figure 9

The figure gives the relation (a) between crossing time ${\tau }^c$ and characteristic size $L$, and (b) between $E_i$ and $L_i$ for the peptides ${\mathrm{GS}}_4$ (black solid square) and ${\mathrm{A}}_8$ (red solid circle) in the subspace spanned with essential components based on xPCA.

Figure 10

The figure gives the relation (a) between crossing time ${\tau }^c$ and characteristic size $L$, and (b) between $E_i$ and $L_i$ for the peptides ${\mathrm{GS}}_4$ (black solid square) and ${\mathrm{A}}_8$ (red solid circle) in the subspace spanned with essential components based on dPCA.

Figure 11

Conformational cluster transition networks (CCTN) for (a) ${\mathrm{GS}}_4$ with 859 vertices and 7691 edges, and (b) ${\mathrm{A}}_8$ with 438 vertices and 1868 edges. The CCTN of ${\mathrm{GS}}_4$ is a uniform network, while that of ${\mathrm{A}}_8$ has some centralized clusters. The normalized degree distributions $P\left(d\right)$ of the CCTN for (c) ${\mathrm{GS}}_4$ and (d) ${\mathrm{A}}_8$ are shown. Lines are fitted by a log-normal distribution. The mean value $\mu$ and the standard deviation $\sigma$ are fit parameters.

Figure 12

The relation between mean square of displacement and the lag time calculated based on topology-only transition probability matrix for the peptides ${\mathrm{GS}}_4$ (open black sqaure) and ${\mathrm{A}}_8$ (open red circle). Local exponents are shown near the curves.