Advances in random matrix theory, zeta functions, and sphere packing

Over four hundred years ago, Sir Walter Raleigh asked his mathematical assistant to find formulas for the number of cannonballs in regularly stacked piles. These investigations aroused the curiosity of the astronomer Johannes Kepler and led to a problem that has gone centuries without a solution: why is the familiar cannonball stack the most efficient arrangement possible? Here we discuss the solution that Hales found in 1998. Almost every part of the 282-page proof relies on long computer verifications. Random matrix theory was developed by physicists to describe the spectra of complex nuclei. In particular, the statistical fluctuations of the eigenvalues (“the energy levels”) follow certain universal laws based on symmetry types. We describe these and then discuss the remarkable appearance of these laws for zeros of the Riemann zeta function (which is the generating function for prime numbers and is the last special function from the last century that is not understood today.) Explaining this phenomenon is a central problem. These topics are distinct, so we present them separately with their own introductory remarks.

Packing at the protein-water interface.

We have determined the packing efficiency at the protein-water interface by calculating the volumes of atoms on the protein surface and nearby water molecules in 22 crystal structures. We find that an atom on the protein surface occupies, on average, a volume approximately 7% larger than an atom of equivalent chemical type in the protein core. In these calculations, larger volumes result from voids between atoms and thus imply a looser or less efficient packing. We further find that the volumes of individual atoms are not related to their chemical type but rather to their structural location. More exposed atoms have larger volumes. Moreover, the packing around atoms in locally concave, grooved regions of protein surfaces is looser than that around atoms in locally convex, ridge regions. This as a direct manifestation of surface curvature-dependent hydration. The net volume increase for atoms on the protein surface is compensated by volume decreases in water molecules near the surface. These waters occupy volumes smaller than those in the bulk solvent by up to 20%; the precise amount of this decrease is directly related to the extent of contact with the protein.

Local densities orthogonal to beta-sheet amide planes: patterns of packing in globular proteins.

We have investigated the efficiency of packing by calculating intramolecular packing density above and below peptide planes of internal beta-pleated sheet residues in five globular proteins. The orientation of interest was chosen to allow study of regions that are approximately perpendicular to the faces of beta-pleated sheets. In these locations, nonbonded van der Waals packing interactions predominate over hydrogen bonding and solvent interactions. We observed considerable variability in packing densities within these regions, confirming that the interior packing of a protein does not result in uniform occupation of the available space. Patterns of fluctuation in packing density suggest that the regular backbone-to-backbone network of hydrogen bonds is not likely to be interrupted to maximize van der Waals interactions. However, high-density packing tends to occur toward the ends of beta-structure strands where hydrogen bonds are more likely to involve nonpolar side-chain groups or solvent molecules. These features result in internal protein folding with a central low-density core surrounded by a higher-density subsurface shell, consistent with our previous calculations regarding overall protein packing density.

A cooperative model for receptor recognition and cell adhesion: evidence from the molecular packing in the 1.6-A crystal structure of the pheromone Er-1 from the ciliated protozoan Euplotes raikovi.

The crystal structure of the pheromone Er-1 from the unicellular eukaryotic organism Euplotes raikovi was determined at 1.6 A resolution and refined to a crystallographic R factor of 19.9%. In the tightly packed crystal, two extensive intermolecular helix-helix interactions arrange the Er-1 molecules into layers. Since the putative receptor of the pheromone is a membrane-bound protein, whose extracellular C-terminal domain is identical in amino acid sequence to the soluble pheromone, the interactions found in the crystal may mimic the pheromone-receptor interactions as they occur on a cell surface. Based on this, we propose a model for the interaction between soluble pheromone molecules and their receptors. In this model, strong pheromone-receptor binding emerges as a consequence of the cooperative utilization of several weak interactions. The model offers an explanation for the results of binding studies and may also explain the adhesion between cells that occurs during mating.

Helix packing of lactose permease in Escherichia coli studied by site-directed chemical cleavage.

Biotinylated lactose permease from Escherichia coli containing a single-cysteine residue at position 330 (helix X) or at position 147, 148, or 149 (helix V) was purified by avidin-affinity chromatography and derivatized with 5-(alpha-bromoacetamido)-1,10-phenanthroline-copper [OP(Cu)]. Studies with purified, OP(Cu)-labeled Leu-330 --> Cys permease in dodecyl-beta-D-maltopyranoside demonstrate that after incubation in the presence of ascorbate, cleavage products of approximately 19 and 6-8 kDa are observed on immunoblots with anti-C-terminal antibody. Remarkably, the same cleavage products are observed with permease embedded in the native membrane. Comparison with the C-terminal half of the permease expressed independently as a standard indicates that the 19-kDa product results from cleavage near the cytoplasmic end of helix VII, whereas the 6- to 8-kDa fragment probably results from fragmentation near the cytoplasmic end of helix XI. Results are entirely consistent with a tertiary-structure model of the C-terminal half of the permease derived from earlier site-directed fluorescence and site-directed mutagenesis studies. Similar studies with OP(Cu)-labeled Cys-148 permease exhibit cleavage products at approximately 19 kDa and at 15-16 kDa. The larger fragment probably reflects cleavage at a site near the cytoplasmic end of helix VII, whereas the 15- to 16-kDa fragment is consistent with cleavage near the cytoplasmic end of helix VIII. When OP(Cu) is moved 100 degrees to position 149 (Val-149 --> Cys permease), a single product is observed at 19 kDa, suggesting fragmentation at the cytoplasmic end of helix VII. However, when the reagent is moved 100 degrees in the other direction to position 147 (Gly-147 --> Cys permease), cleavage is not observed. The results suggest that helix V is in close proximity to helices VII and VIII with position 148 in the interface between the helices, position 149 facing helix VII, and position 147 facing the lipid bilayer.

Bill of lading for shipment of four boxes of books, Oct. 20, 1764

This bill of lading is from Jasper Mauduit's shipment of books (his own gift and those bequeathed by Dr. Avery) to Andrew Oliver. The books were shipped from London to Boston via a ship called Hale Galley, under the direction of Captain Harris Hatch.