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.

The vibrational energy flow transition in organic molecules: Theory meets experiment

Most large dynamical systems are thought to have ergodic dynamics, whereas small systems may not have free interchange of energy between degrees of freedom. This assumption is made in many areas of chemistry and physics, ranging from nuclei to reacting molecules and on to quantum dots. We examine the transition to facile vibrational energy flow in a large set of organic molecules as molecular size is increased. Both analytical and computational results based on local random matrix models describe the transition to unrestricted vibrational energy flow in these molecules. In particular, the models connect the number of states participating in intramolecular energy flow to simple molecular properties such as the molecular size and the distribution of vibrational frequencies. The transition itself is governed by a local anharmonic coupling strength and a local state density. The theoretical results for the transition characteristics compare well with those implied by experimental measurements using IR fluorescence spectroscopy of dilution factors reported by Stewart and McDonald [Stewart, G. M. & McDonald, J. D. (1983) J. Chem. Phys. 78, 3907–3915].

Interaction of voltage-gated sodium channels with the extracellular matrix molecules tenascin-C and tenascin-R

The type IIA rat brain sodium channel is composed of three subunits: a large pore-forming α subunit and two smaller auxiliary subunits, β1 and β2. The β subunits are single membrane-spanning glycoproteins with one Ig-like motif in their extracellular domains. The Ig motif of the β2 subunit has close structural similarity to one of the six Ig motifs in the extracellular domain of the cell adhesion molecule contactin (also called F3 or F11), which binds to the extracellular matrix molecules tenascin-C and tenascin-R. We investigated the binding of the purified sodium channel and the extracellular domain of the β2 subunit to tenascin-C and tenascin-R in vitro. Incubation of purified sodium channels on microtiter plates coated with tenascin-C revealed saturable and specific binding with an apparent Kd of ≈15 nM. Glutathione S-transferase-tagged fusion proteins containing various segments of tenascin-C and tenascin-R were purified, digested with thrombin to remove the epitope tag, immobilized on microtiter dishes, and tested for their ability to bind purified sodium channel or the epitope-tagged extracellular domain of β2 subunits. Both purified sodium channels and the extracellular domain of the β2 subunit bound specifically to fibronectin type III repeats 1–2, A, B, and 6–8 of tenascin-C and fibronectin type III repeats 1–2 and 6–8 of tenascin-R but not to the epidermal growth factor-like domain or the fibrinogen-like domain of these molecules. The binding of neuronal sodium channels to extracellular matrix molecules such as tenascin-C and tenascin-R may play a crucial role in localizing sodium channels in high density at axon initial segments and nodes of Ranvier or in regulating the activity of immobilized sodium channels in these locations.