How Complex Is the Dynamics of Peptide Folding?

Classical molecular dynamics simulations of the folding of alanine peptides in aqueous solution are analyzed by constructing a deterministic model of the dynamics, using methods from nonlinear time series analysis. While the dimension of the free energy landscape increases with system size, a Lyapunov analysis shows that the effective dimension of the dynamic system is rather small and even decreases with chain length. The observed reduction of phase space is a nonlinear cooperative effect that is caused by intramolecular hydrogen bonds that stabilize the secondary structure of the peptides.

Figure 1

MD snapshots of (left) an extended conformation of ${\mathrm{Ala}}_3$ showing the central backbone dihedral angles $\varphi$ and $\psi$, and (right) the ${\alpha }_R$ helix conformation of ${\mathrm{Ala}}_{10}$ indicating the stabilizing $n-(n+4)$ hydrogen bonds.

Figure 2

Time series $v_i(t)$, distributions $P(v_i)$, and autocorrelation functions $C_i(t)$ obtained for the first two principal components of the ${\mathrm{Ala}}_3$ system. The solid black lines represent the results of the MD simulation, the dashed red (or gray) lines correspond to results from the nonlinear model of the dynamics.

Figure 3

Same as in Fig. 2, but for the ${\mathrm{Ala}}_{10}$ system.