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.

Crystal structure of human p32, a doughnut-shaped acidic mitochondrial matrix protein

Human p32 (also known as SF2-associated p32, p32/TAP, and gC1qR) is a conserved eukaryotic protein that localizes predominantly in the mitochondrial matrix. It is thought to be involved in mitochondrial oxidative phosphorylation and in nucleus–mitochondrion interactions. We report the crystal structure of p32 determined at 2.25 Å resolution. The structure reveals that p32 adopts a novel fold with seven consecutive antiparallel β-strands flanked by one N-terminal and two C-terminal α-helices. Three monomers form a doughnut-shaped quaternary structure with an unusually asymmetric charge distribution on the surface. The implications of the structure on previously proposed functions of p32 are discussed and new specific functional properties are suggested.

Systematic derivation of partition functions for ligand binding to two-dimensional lattices

The Ising problem consists in finding the analytical solution of the partition function of a lattice once the interaction geometry among its elements is specified. No general analytical solution is available for this problem, except for the one-dimensional case. Using site-specific thermodynamics, it is shown that the partition function for ligand binding to a two-dimensional lattice can be obtained from those of one-dimensional lattices with known solution. The complexity of the lattice is reduced recursively by application of a contact transformation that involves a relatively small number of steps. The transformation implemented in a computer code solves the partition function of the lattice by operating on the connectivity matrix of the graph associated with it. This provides a powerful new approach to the Ising problem, and enables a systematic analysis of two-dimensional lattices that model many biologically relevant phenomena. Application of this approach to finite two-dimensional lattices with positive cooperativity indicates that the binding capacity per site diverges as Na (N = number of sites in the lattice) and experiences a phase-transition-like discontinuity in the thermodynamic limit N → ∞. The zeroes of the partition function tend to distribute on a slightly distorted unit circle in complex plane and approach the positive real axis already for a 5×5 square lattice. When the lattice has negative cooperativity, its properties mimic those of a system composed of two classes of independent sites with the apparent population of low-affinity binding sites increasing with the size of the lattice, thereby accounting for a phenomenon encountered in many ligand-receptor interactions.

Assembly of a Novel Cartilage Matrix Protein Filamentous Network: Molecular Basis of Differential Requirement of von Willebrand Factor A Domains

Cartilage matrix protein (CMP) is the prototype of the newly discovered matrilin family, all of which contain von Willebrand factor A domains. Although the function of matrilins remain unclear, we have shown that, in primary chondrocyte cultures, CMP (matrilin-1) forms a filamentous network, which is made up of two types of filaments, a collagen-dependent one and a collagen-independent one. In this study, we demonstrate that the collagen-independent CMP filaments are enriched in pericellular compartments, extending directly from chondrocyte membranes. Their morphology can be distinguished from that of collagen filaments by immunogold electron microscopy, and mimicked by that of self-assembled purified CMP. The assembly of CMP filaments can occur from transfection of a wild-type CMP transgene alone in skin fibroblasts, which do not produce endogenous CMP. Conversely, assembly of endogenous CMP filaments by chondrocytes can be inhibited specifically by dominant negative CMP transgenes. The two A domains within CMP serve essential but different functions during network formation. Deletion of the A2 domain converts the trimeric CMP into a mixture of monomers, dimers, and trimers, whereas deletion of the A1 domain does not affect the trimeric configuration. This suggests that the A2 domain modulates multimerization of CMP. Absence of either A domain from CMP abolishes its ability to form collagen-independent filaments. In particular, Asp22 in A1 and Asp255 in A2 are essential; double point mutation of these residues disrupts CMP network formation. These residues are part of the metal ion–dependent adhesion sites, thus a metal ion–dependent adhesion site–mediated adhesion mechanism may be applicable to matrilin assembly. Taken together, our data suggest that CMP is a bridging molecule that connects matrix components in cartilage to form an integrated matrix network.

Functional Hierarchy of Simultaneously Expressed Adhesion Receptors: Integrin α2β1 but Not CD44 Mediates MV3 Melanoma Cell Migration and Matrix Reorganization within Three-dimensional Hyaluronan-containing Collagen MatricesV⃞

Haptokinetic cell migration across surfaces is mediated by adhesion receptors including β1 integrins and CD44 providing adhesion to extracellular matrix (ECM) ligands such as collagen and hyaluronan (HA), respectively. Little is known, however, about how such different receptor systems synergize for cell migration through three-dimensionally (3-D) interconnected ECM ligands. In highly motile human MV3 melanoma cells, both β1 integrins and CD44 are abundantly expressed, support migration across collagen and HA, respectively, and are deposited upon migration, whereas only β1 integrins but not CD44 redistribute to focal adhesions. In 3-D collagen lattices in the presence or absence of HA and cross-linking chondroitin sulfate, MV3 cell migration and associated functions such as polarization and matrix reorganization were blocked by anti-β1 and anti-α2 integrin mAbs, whereas mAbs blocking CD44, α3, α5, α6, or αv integrins showed no effect. With use of highly sensitive time-lapse videomicroscopy and computer-assisted cell tracking techniques, promigratory functions of CD44 were excluded. 1) Addition of HA did not increase the migratory cell population or its migration velocity, 2) blocking of the HA-binding Hermes-1 epitope did not affect migration, and 3) impaired migration after blocking or activation of β1 integrins was not restored via CD44. Because α2β1-mediated migration was neither synergized nor replaced by CD44–HA interactions, we conclude that the biophysical properties of 3-D multicomponent ECM impose more restricted molecular functions of adhesion receptors, thereby differing from haptokinetic migration across surfaces.

Apyrase Functions in Plant Phosphate Nutrition and Mobilizes Phosphate from Extracellular ATP1

ATP, which is present in the extracellular matrix of multicellular organisms and in the extracellular fluid of unicellular organisms, has been shown to function as a signaling molecule in animals. The concentration of extracellular ATP (xATP) is known to be functionally modulated in part by ectoapyrases, membrane-associated proteins that cleave the γ- and β-phosphates on xATP. We present data showing a previously unreported (to our knowledge) linkage between apyrase and phosphate transport. An apyrase from pea (Pisum sativum) complements a yeast (Saccharomyces cerevisiae) phosphate-transport mutant and significantly increases the amount of phosphate taken up by transgenic plants overexpressing the gene. The transgenic plants show enhanced growth and augmented phosphate transport when the additional phosphate is supplied as inorganic phosphate or as ATP. When scavenging phosphate from xATP, apyrase mobilizes the γ-phosphate without promoting the transport of the purine or the ribose.

Crystal structures of the trimeric human immunodeficiency virus type 1 matrix protein: implications for membrane association and assembly.

The human immunodeficiency virus type 1 (HIV-1) matrix protein forms a structural shell associated with the inner viral membrane and performs other essential functions throughout the viral life cycle. The crystal structure of the HIV-1 matrix protein, determined at 2.3 angstrom resolution, reveals that individual matrix molecules are composed of five major helices capped by a three-stranded mixed beta-sheet. Unexpectedly, the protein assembles into a trimer in three different crystal lattices, burying 1880 angstrom2 of accessible surface area at the trimer interfaces. Trimerization appears to create a large, bipartite membrane binding surface in which exposed basic residues could cooperate with the N-terminal myristoyl groups to anchor the protein on the acidic inner membrane of the virus.