Subdiffusion in Peptides Originates from the Fractal-Like Structure of Configuration Space

Molecular dynamics simulation of oligopeptide chains reveals configurational subdiffusion at equilibrium extending from ${10}^{-12}$ to ${10}^{-8}\mathrm{s}$. Trap models, involving a random walk with a distribution of waiting times, cannot account for the subdiffusion, which is found rather to arise from the fractal-like structure of the accessible configuration space.

Figure 1

(a) MSD for different PCs of $(GS{)}_{5}W$. (b) Mean squared displacement exponents $\alpha$ for the four peptides as a function of mode number, obtained by fitting a linear function of $\log t$ to $\log ⟨x(t{)}^2⟩$ in the $t$ range 1 to 10 ps using a least-squares method. Error bars are smaller than symbol size.

Figure 2

Waiting-time distribution for different PCs of $(GS{)}_{5}W$ obtained by projecting time series of each PC onto a two-state space. For PCs with multiminimum potentials of mean force, i.e., the lowest, the two states were defined by dividing the $x$ space into the two parts either side of the position of the highest barrier in the potential of the mean force. For the higher PCs, which have only a single minimum, by symmetry the space was divided at the position of the minimum. The results were found to be independent of the precise location of the partition. In order to improve statistics at long times, the data were piecewise convoluted with filters of different sizes.

Figure 3

Normalized MSD of a CTRW with $\beta =0.5$. The waiting times $t_w$ were generated from a uniform random variable $r\in [0,1]$ using the transformation $t_w=r^{-1/\beta }-1$. Jump lengths were taken from a normal distribution. The red line (“ens“) shows $⟨x^2(t)⟩$ calculated as an ensemble average over 10 000 realizations of a CTRW. $⟨x^2(t)⟩$ calculated as a time averaged quantity is given in magenta (“time“). The blue and green lines (dashed: see legend) give the asymptotic power-law behavior and serve to guide the eye.

Figure 4

MSD of trajectories generated from the transition matrix $R(\Delta t)$ for different $\Delta t$ together with the corresponding MSD from the original trajectory. The matrix $R(\Delta t)$ was derived from the $(GS{)}_{5}W$ trajectory in the subspace of the first ten PCs.