Hydration dependence of the mass fractal dimension and anomalous diffusion of vibrational energy in proteins

Vibrational dynamics of proteins and energy flow depend on protein geometry as well as interactions of a protein molecule with the surrounding solvent. We compute the mass fractal dimension $D$ of proteins ranging from 100 to over 10 000 amino acids comparing values for the bare protein with those computed when buried and hydration waters are included in the calculation. Including water in the calculation increases $D$ by about 0.3 to 2.87 on average above $D$ computed for the dehydrated protein. The mass fractal dimension of proteins that are partially unfolded by molecular dynamics (MD) simulation is also computed and found to vary little when the radius of gyration changes within about 10% of that for the Protein Data Bank structure. MD simulations of vibrational energy diffusion in proteins reveal that the exponent characterizing anomalous diffusion of vibrational energy does not change much with hydration, which is seen to be due to an increase in the spectral dimension with hydration by a factor similar to the increase in $D$.

Figure 1

Plot of ${\log }_{10}M$ vs ${\log }_{10}R$ for a box of water molecules (points connected by solid line) and the protein 1 CW3 (two sets of points connected by dashed lines), where values of $M$ are the masses enclosed by concentric spheres of radius $R$ centered at an O atom for water, or a $C_{\alpha }$ for the protein. The two sets of points for the protein correspond to using the closest $C_{\alpha }$ to the center of mass of the protein. Points connected by the long-dashed line correspond to a calculation of $M$ including water molecules, whereas those connected by the short-dashed line correspond to a calculation of $M$ where only protein atoms are included.

Figure 2

Values of $D_{10\%}$ (gray) and $D$ (black) computed for 200 proteins in the PDB, each containing $N$ residues. Calculations were carried out by excluding water molecules, in which case the dimension is the value at the bottom of the vertical line in the plot, and including the water in the PDB, in which case the dimension is the value at the top of the vertical line.

Figure 3

Same as Fig. 2, but only for the 20 proteins listed in Table . Values of $D$ were computed after $50\text{−}\mathrm{ps}$ MD simulations. The value at the top of the vertical bars represents the dimension computed with inclusion of the surrounding water molecules.

Figure 4

(a) The fraction of water mass to total mass contained in spheres of radius $R$ (Å), around the 10% of $C_{\alpha }$’s closest to the center of mass of the proteins listed in Table is plotted against protein size $N$ and radius $R$; (b) shows this fraction for some of these proteins as a function of $R$. Values of $N$ plotted are 159 (filled circles), 249 (open circles), 258 (filled squares), 360 (open squares), 474 (filled triangles), 534 (open triangles), 683 (filled diamonds), 988 (open diamonds), 1720 (×), and 2462 (+).

Figure 5

Plot of (a) $D_{10\%}$ and (b) $D$ for the ten smallest proteins listed in Table , as a function of $R_G$ for the native structure, relative to $R_G$ of a particular structure obtained by partial unfolding during MD simulation, $R_G\left(\text{native}\right)∕R_G$. $N$ is 104 (filled circles), 159 (filled squares), 214 (filled triangles), 225 (filled diamonds), 258 (open circles), 309 (open squares), 322 (open triangles), 360 (open diamonds), 401 (×), and 474 (+).

Figure 6

Plot of $\log [⟨R^2\left(t\right)⟩-⟨R^2\left(0\right)⟩]$ vs $\log \left(t\right)$ for green fluorescent protein (1 emb), without water (diamonds) and with the 600 water molecules (×) within $3\mathrm{\AA }$ of the protein. The slopes of the plotted lines fit to the results from $0.1\text{to}2.0\mathrm{ps}$ are 0.60 and 0.61 for the dehydrated and hydrated protein, respectively, indicating anomalous subdiffusion with exponent $\nu =0.30$ for both systems.

Figure 7

The log of the vibrational density, ${\log }_{10}\rho$, is plotted vs the log of the vibrational frequency, ${\log }_{10}\omega$, for 1 emb from frequencies of $5–60{\mathrm{cm}}^{-1}$. The slope of the line through the data is $\overline{d}-1$, where $\overline{d}$ is the spectral dimension. A best fit through the points for 1 emb without (diamonds) and with (×) hydration water gives spectral dimensions of 1.45 and 1.74, respectively.