Figure 1

Calculating the spectral dimension $d_s$ for PDB codes: 1V97 ($N=2594$, $d_s=1.78$), 1E7U ($N=872$, $d_s=1.56$), and 1VPD ($N=279$, $d_s=1.49$). For each protein, we found the set of vibrational eigenfrequencies $\{{\omega }_0,{\omega }_1,\dots ,{\omega }_{N-1}\}$ that characterize the elastic network it forms when modeled by the GNM and plotted $\ln [G(\omega )]$ vs $\ln (\omega )$. In this example $R_c=6\AA$ and $G(\omega )$ is the cumulative density of modes defined as $G(\omega )={\int }_0^{\omega }g({\omega }^{\prime })d{\omega }^{\prime }$. All obtained modes are shown. The low frequency regions of $G(\omega )$ clearly exhibit a power law behavior; i.e., the scaling relation $G(\omega )\sim {\omega }^{d_s}$ holds for low frequencies. Dashed lines indicate best fits to these regions, $\omega \in [0.109,1.67]$, $\omega \in [0.119,1.6]$, and $\omega \in [0.243,0.888]$ for 1V97, 1E7U, and 1VPD, correspondingly; the slopes correspond to the spectral dimensions. Inset: Calculating the mass fractal dimension $d_f$ for PDB code 1V97 ($N=2594$, $d_f=2.64$), $d_f$ was taken to be the average mass fractal dimension obtained by choosing the origin to be each and every one of the ten $C\mathrm{\text{−}}\alpha$ atoms closest to the protein’s center of mass. For a given origin, $d_f$ was estimated via a power law fitting to $M(r)$, dashed lines indicate best fits. The data points as well as the best fits for different origins overlap significantly; we take the average slope to be the fractal dimension.Reuse & Permissions

Figure 2

Fitting the data gathered for 543 proteins with the equation $\frac{2}{d_s}+\frac{1}{d_f}=a+\frac{b}{\ln (N)}$. Here, the spectral dimension was calculated for $R_c=6\AA$. The best-fit parameters are $a=0.90$ and $b=4.53$, the correlation coefficient is 0.55. Prediction bounds are for a confidence level of 95%.Reuse & Permissions