On the riemann zeta-function and the divisor problem II

Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of . If E ^*(t)=E(t)-2πΔ^*(t/2π) with , then we obtain

The average prime divisor of an integer in short intervals

Archiv der Mathematik -     

Relation between pore sizes of protein crystals and anisotropic solute diffusivities

The diffusion of a solute, fluorescein, into lysozyme protein crystals with different pore structures was investigated. To determine the diffusion coefficients, three-dimensional solute concentration fields acquired by confocal laser scanning microscopy (CLSM) during diffusion into the crystals were compared with the output of a time-dependent 3-D diffusion model. The diffusion process was found to be anisotropic, and the degree of anisotropy increased in the order: triclinic, tetragonal and orthorhombic crystal morphology. A linear correlation between the pore diffusion coefficients and the pore sizes was established. The maximum size of the solute, deduced from the established correlation of diffusion coefficients and pore size, was 0.73 +/- 0.06 nm, which was in the range of the average diameter of fluorescein (0.69 +/- 0.02 nm). This proves that size exclusion is the key mechanism for solute diffusion in protein crystals. Hence, the origin of solute diffusion anisotropy can be found in the packing of the protein molecules in the crystals, which determines the crystal pore organization.     

Adsorption of xanthene dyes by lysozyme crystals

Adsorption characteristics of cross-linked lysozyme crystals of different morphologies (tetragonal, orthorhombic, triclinic and monoclinic) were examined using four anionic dyes (fluorescein, eosin, erythrosin, and rose bengal), one zwitterionic dye (rhodamine B), and one cationic dye (rhodamine 6G). The adsorption isotherms were of the Langmuir type for all examined systems with the exception of rhodamine B adsorption by monoclinic crystals. The weakest adsorption was observed for the cationic dye, rhodamine B, whereas dianionic dyes, eosin, rose bengal, and erythrosin were strongly adsorbed on the protein surface. The adsorption capacities of the crystals for the dyes were found to depend on both charge and hydrophobicity of the dye, reflecting the heterogeneous character of the lysozyme pore surface. The adsorption affinity of the crystals for the dyes was a function of the dyes' hydrophobicity. Furthermore, the crystal morphology was identified as an additional factor determining capacity and affinity for dye adsorption. Differences between crystals prepared in the presence of the same precipitant were lower than between morphologies prepared with different precipitants.