Anharmonicity and Self-Similarity of the Free Energy Landscape of Protein $G$

The near-native free-energy landscape of protein $G$ is investigated through $0.4–\mu \mathrm{s}$-long atomistic molecular dynamics simulations in an explicit solvent. A theoretical and computational framework is used to assess the time dependence of salient thermodynamical features. While the quasiharmonic character of the free energy is found to degrade in a few ns, the slow modes display a very mild dependence on the trajectory duration. This property originates from a striking self-similarity of the free-energy landscape embodied by the consistency of the principal directions of the local minima, where the system dwells for several ns, and of the virtual jumps connecting them.

Figure 1

(a) Ribbon representation of protein $G$. (b) Color-coded plot of the Kendall’s correlation coefficient, $\tau$, between corresponding elements of the coupling matrices, $M$, calculated over the first $\Delta t$ and $\Delta t^{\prime }$ ns of the first trajectory. Matrix elements along the diagonal or pertaining to consecutive $C_{\alpha }$’s were omitted in the calculation of $\tau$.

Figure 2

Normalized distribution of the projection along the first slow mode of the conformational fluctuations encountered in intervals of increasing length from the first trajectory.

Figure 3

Top: time evolution of the projection along the 1st eigenvector and of the variational deterministic oscillator for two intervals of different duration. Bottom: average value of the variational oscillator frequency as a function of the interval duration, $\Delta t$.

Figure 4

(a) Histogram of the RMSIP values calculated for the principal components of all pairs of top structural clusters. (b) Histogram of the norm of the projection of all pairwise distance vectors between cluster representatives along the top 10 principal components of the largest cluster.

Figure 5

RMSIP between the top essential dynamical spaces of the 1st ns of trajectory 1 and intervals of longer duration, $\Delta t$, from the same and other trajectories.