Mass fractal dimension and the compactness of proteins

Vibrational dynamics and energy flow in a protein are related by Alexander-Orbach theory to the protein’s mass fractal dimension $D$ and spectral dimension $\overline{d}$. Burioni et al. [Proteins: Struct., Funct. Bioinf. 55, 529 (2004)] recently proposed a relation between $\overline{d}$ and protein size based on their computational analysis of a set of proteins ranging from about 100 to several thousand amino acids. We report here values for $D$ computed for 200 proteins from the Protein Data Bank (PDB) ranging from about 100 to over 10 000 amino acids and examine variation of $D$ with protein size. The average $D$ is found to be 2.5, significantly smaller than a completely compact three-dimensional collapsed polymer. Indeed, we find that on average a protein in its PDB configuration fills about three-quarters of the volume within the protein surface. Protein mass is also found to scale with radius of gyration with an exponent of 2.5 for this set of proteins.

Figure 1

Plot of ${\log }_{10}M$ vs ${\log }_{10}R$ for 1MZ5, where values of $M$ are the masses enclosed by concentric spheres of radius $R$ centered at a backbone atom. Each of the ten sets of points through which lines are fitted corresponds to a center in our calculation, which is one of the ten closest ${\mathrm{C}}_{\alpha }$’s to the center of mass of the protein. The correlation coefficients for the lines that best fit the data range from 0.9985 to 0.9997.

Figure 2

(a) Average values of the mass fractal dimension $D$ computed for the 200 proteins using as centers in the calculation the nearest 10%, 10–20 %, 20–30 %, etc., of the ${\mathrm{C}}_{\alpha }$’s from the center of mass of the protein (light gray). Also shown is $D$ computed using all ${\mathrm{C}}_{\alpha }$’s as centers (dark gray). (b) Same as (a), but for the 63 proteins with 200–400 amino acids. (c) Same as (a), but for the ten largest proteins in the data set.

Figure 3

Plot of the mass fractal dimension $D$ (squares) and its estimate using as centers in the calculation the nearest 10% of all ${\mathrm{C}}_{\alpha }$’s to the center of mass of the protein, $D_{10\%}$ (triangles), as a function of the number of amino acids of each protein, $N$. For this set of 200 proteins we find $D$ is $2.489\pm 0.172$ and $D_{10\%}$ is $2.761\pm 0.164$.

Figure 4

Plot of ${\log }_{10}{R}_G$ vs ${\log }_{10}N$ for the 200 proteins in the set. Data are labeled as × for a protein with lower-than-average $D$, i.e., $D<2.489$; and 엯 for a protein with $D>2.489$. The best-fit line, with correlation coefficient 0.9893, is drawn through the data. The slope of this line, which can be interpreted as $1∕D$, is 0.390, giving an estimate of 2.56 for the dimension.

Figure 5

Cross section of the protein 1A4G superposed on a lattice of $1\text{−}\mathrm{\AA }$ cells, as described in text. White cells are computed to lie outside the protein surface. Black cells contain a protein atom and dark gray cells contain part of the volume of a protein atom. Light gray cells represent the empty or void spaces within the protein surface. The cross section in (a) has been computed with the algorithm described in the text. In (b) and (c) we remove the outer layer of void cells, which are an artifact of our computation of the protein surface and are removed to compute the fraction of void space $f_V$ within the protein surface. In (b) and (c) we use in our computation a sphere of 1.5 and $2.0\mathrm{\AA }$, respectively, for each protein atom.

Figure 6

Plot of the void fraction $f_V$ calculated using as radius for each protein atom a value of 1.5 (triangles) and 2.0 (squares) Å, as a function of the number of amino acids of each protein, $N$. For this set of 200 proteins we find $f_V$ is $0.265\pm 0.035$ using the smaller radius and is $0.139\pm 0.030$ with the larger.