Estimate for the fractal dimension of the Apollonian gasket in d dimensions

Revised scattering exponents for a power-law distribution of surface and mass fractals

We consider scattering exponents arising in small-angle scattering from power-law polydisperse surface and mass fractals. It is shown that a set of fractals, whose sizes are distributed according to a power law, can change its fractal dimension when the power-law exponent is sufficiently big. As a result, the scattering exponent corresponding to this dimension appears due to the spatial correlations between positions of different fractals. For large values of the momentum transfer, the correlations do not play any role, and the resulting scattering intensity is given by a sum of intensities of all composing fractals. The restrictions imposed on the power-law exponents are found. The obtained results generalize Martin's formulas for the scattering exponents of the polydisperse fractals.

Packing in protein cores

Proteins are biological polymers that underlie all cellular functions. The first high-resolution protein structures were determined by x-ray crystallography in the 1960s. Since then, there has been continued interest in understanding and predicting protein structure and stability. It is well-established that a large contribution to protein stability originates from the sequestration from solvent of hydrophobic residues in the protein core. How are such hydrophobic residues arranged in the core; how can one best model the packing of these residues, and are residues loosely packed with multiple allowed side chain conformations or densely packed with a single allowed side chain conformation? Here we show that to properly model the packing of residues in protein cores it is essential that amino acids are represented by appropriately calibrated atom sizes, and that hydrogen atoms are explicitly included. We show that protein cores possess a packing fraction of [Formula: see text], which is significantly less than the typically quoted value of 0.74 obtained using the extended atom representation. We also compare the results for the packing of amino acids in protein cores to results obtained for jammed packings from discrete element simulations of spheres, elongated particles, and composite particles with bumpy surfaces. We show that amino acids in protein cores pack as densely as disordered jammed packings of particles with similar values for the aspect ratio and bumpiness as found for amino acids. Knowing the structural properties of protein cores is of both fundamental and practical importance. Practically, it enables the assessment of changes in the structure and stability of proteins arising from amino acid mutations (such as those identified as a result of the massive human genome sequencing efforts) and the design of new folded, stable proteins and protein-protein interactions with tunable specificity and affinity.