Advillin Folding Takes Place on a Hypersurface of Small Dimensionality

All-atom explicit-solvent molecular dynamics simulations have been used to investigate the topological structure of the space explored during folding by the $c$-terminal fragment of the Advillin headpiece, a 36 amino-acid protein. A fractal dimension analysis shows that the hypersurface explored during the folding process has an approximate dimensionality of only three. It is shown that this low dimensionality persists well above the unfolding temperature and is not present in simple coarse-grained models.

Figure 1

Plot of the logarithm of the number of neighbors $N$ as a function of the logarithm of the distance $r$. Since $r$ has to be defined in a space including variables with different units of measure, before computing the distance each variable $s$ is divided by $s_{\max }-s_{\min }$, where $s_{\max }$ and $s_{\min }$ are the maximum and the minimum value of $s$ explored during the dynamics. Data are reported for simulation 1 (green), simulation 2 (red), half of the data of simulation 2 (blue), simulation 2 in a 30-dimensional representation based on the number of $C\alpha$ contacts of residues 4–33 (black) and simulation 2 in a 15-dimensional space based on the $C\alpha \mathrm{\text{−}}C\alpha$ distances (purple). The black line corresponds to a correlation dimension of 3 and is plotted as a guide for the eyes. For the sake of clarity, the curves obtained in the different simulations are displaced by an arbitrary constant in $y$ direction, otherwise some of them would be indistinguishable.

Figure 2

Temperature dependence of the correlation dimension of the folding space. The correlation dimension was estimated as in Fig. 1 for each replica of a parallel tempering simulation. The dashed line indicates the melting temperature (342 K).

Figure 3

The logarithm of the number of neighbors $N$ as a function of the distance $r$ in logarithmic scale for a Go model simulations. $r$ is computed in the 15-dimensional space based on the $C\alpha \mathrm{\text{−}}C\alpha$ distances. The correlation dimension was calculated at a temperature where the protein is approximately 50% folded on a trajectory containing hundreds of folding events (× crosses); same as above, but on a five-times shorter trajectory (pluses); for a purely repulsive self-avoiding polymer (stars). The curve computed with the same set of variables for the all-atom simulation is reported in Fig. 1, purple.