Architecture and mechanism of the light-harvesting apparatus of purple bacteria

Photosynthetic organisms fuel their metabolism with light energy and have developed for this purpose an efficient apparatus for harvesting sunlight. The atomic structure of the apparatus, as it evolved in purple bacteria, has been constructed through a combination of x-ray crystallography, electron microscopy, and modeling. The detailed structure and overall architecture reveals a hierarchical aggregate of pigments that utilizes, as shown through femtosecond spectroscopy and quantum physics, elegant and efficient mechanisms for primary light absorption and transfer of electronic excitation toward the photosynthetic reaction center.

Computer simulations of enzyme catalysis: Finding out what has been optimized by evolution

The origin of the catalytic power of enzymes is discussed, paying attention to evolutionary constraints. It is pointed out that enzyme catalysis reflects energy contributions that cannot be determined uniquely by current experimental approaches without augmenting the analysis by computer simulation studies. The use of energy considerations and computer simulations allows one to exclude many of the popular proposals for the way enzymes work. It appears that the standard approaches used by organic chemists to catalyze reactions in solutions are not used by enzymes. This point is illustrated by considering the desolvation hypothesis and showing that it cannot account for a large increase in kcat relative to the corresponding kcage for the reference reaction in a solvent cage. The problems associated with other frequently invoked mechanisms also are outlined. Furthermore, it is pointed out that mutation studies are inconsistent with ground state destabilization mechanisms. After considering factors that were not optimized by evolution, we review computer simulation studies that reproduced the overall catalytic effect of different enzymes. These studies pointed toward electrostatic effects as the most important catalytic contributions. The nature of this electrostatic stabilization mechanism is far from being obvious because the electrostatic interaction between the reacting system and the surrounding area is similar in enzymes and in solution. However, the difference is that enzymes have a preorganized dipolar environment that does not have to pay the reorganization energy for stabilizing the relevant transition states. Apparently, the catalytic power of enzymes is stored in their folding energy in the form of the preorganized polar environment.

Euler characteristics and elliptic curves

Let E be a modular elliptic curve over ℚ, without complex multiplication; let p be a prime number where E has good ordinary reduction; and let F∞ be the field obtained by adjoining to ℚ all p-power division points on E. Write G∞ for the Galois group of F∞ over ℚ. Assume that the complex L-series of E over ℚ does not vanish at s = 1. If p ⩾ 5, we make a precise conjecture about the value of the G∞-Euler characteristic of the Selmer group of E over F∞. If one makes a standard conjecture about the behavior of this Selmer group as a module over the Iwasawa algebra, we are able to prove our conjecture. The crucial local calculations in the proof depend on recent joint work of the first author with R. Greenberg.

p-adic L functions and trivial zeroes

The following is adapted from the notes for the lecture. It announces results and conjectures about values of the p-adic L function of the symmetric square of an elliptic curve.

Adjoint modular Galois representations and their Selmer groups

In the last 15 years, many class number formulas and main conjectures have been proven. Here, we discuss such formulas on the Selmer groups of the three-dimensional adjoint representation ad(φ) of a two-dimensional modular Galois representation φ. We start with the p-adic Galois representation φ0 of a modular elliptic curve E and present a formula expressing in terms of L(1, ad(φ0)) the intersection number of the elliptic curve E and the complementary abelian variety inside the Jacobian of the modular curve. Then we explain how one can deduce a formula for the order of the Selmer group Sel(ad(φ0)) from the proof of Wiles of the Shimura–Taniyama conjecture. After that, we generalize the formula in an Iwasawa theoretic setting of one and two variables. Here the first variable, T, is the weight variable of the universal p-ordinary Hecke algebra, and the second variable is the cyclotomic variable S. In the one-variable case, we let φ denote the p-ordinary Galois representation with values in GL2(Zp[[T]]) lifting φ0, and the characteristic power series of the Selmer group Sel(ad(φ)) is given by a p-adic L-function interpolating L(1, ad(φk)) for weight k + 2 specialization φk of φ. In the two-variable case, we state a main conjecture on the characteristic power series in Zp[[T, S]] of Sel(ad(φ) ⊗ ν−1), where ν is the universal cyclotomic character with values in Zp[[S]]. Finally, we describe our recent results toward the proof of the conjecture and a possible strategy of proving the main conjecture using p-adic Siegel modular forms.

The structure of Selmer groups

The purpose of this article is to describe certain results and conjectures concerning the structure of Galois cohomology groups and Selmer groups, especially for abelian varieties. These results are analogues of a classical theorem of Iwasawa. We formulate a very general version of the Weak Leopoldt Conjecture. One consequence of this conjecture is the nonexistence of proper Λ-submodules of finite index in a certain Galois cohomology group. Under certain hypotheses, one can prove the nonexistence of proper Λ-submodules of finite index in Selmer groups. An example shows that some hypotheses are needed.

On the coefficients of the characteristic series of the U-operator

A conceptual proof is given of the fact that the coefficients of the characteristic series of the U-operator acting on families of overconvegent modular forms lie in the Iwasawa algebra.

Zeta functions and Eisenstein series on classical groups

We construct an Euler product from the Hecke eigenvalues of an automorphic form on a classical group and prove its analytic continuation to the whole complex plane when the group is a unitary group over a CM field and the eigenform is holomorphic. We also prove analytic continuation of an Eisenstein series on another unitary group, containing the group just mentioned defined with such an eigenform. As an application of our methods, we prove an explicit class number formula for a totally definite hermitian form over a CM field.

Deforming semistable Galois representations

Let V be a p-adic representation of Gal(Q̄/Q). One of the ideas of Wiles’s proof of FLT is that, if V is the representation associated to a suitable autromorphic form (a modular form in his case) and if V′ is another p-adic representation of Gal(Q̄/Q) “closed enough” to V, then V′ is also associated to an automorphic form. In this paper we discuss which kind of local condition at p one should require on V and V′ in order to be able to extend this part of Wiles’s methods.

Integrality of Tate-cycles

We explain a technical result about p-adic cohomology and apply it to the study of Shimura varieties. The technical result applies to algebraic varieties with torsion-free cohomology, but for simplicity we only treat abelian varieties.

Congruences between modular forms: Raising the level and dropping Euler factors

We discuss the relationship among certain generalizations of results of Hida, Ribet, and Wiles on congruences between modular forms. Hida’s result accounts for congruences in terms of the value of an L-function, and Ribet’s result is related to the behavior of the period that appears there. Wiles’ theory leads to a class number formula relating the value of the L-function to the size of a Galois cohomology group. The behavior of the period is used to deduce that a formula at “nonminimal level” is obtained from one at “minimal level” by dropping Euler factors from the L-function.

On degree 2 Galois representations over

We discuss proofs of some new special cases of Serre’s conjecture on odd, degree 2 representations of Gℚ.

Molecular kinesis in cellular function and plasticity

Intracellular transport and localization of cellular components are essential for the functional organization and plasticity of eukaryotic cells. Although the elucidation of protein transport mechanisms has made impressive progress in recent years, intracellular transport of RNA remains less well understood. The National Academy of Sciences Colloquium on Molecular Kinesis in Cellular Function and Plasticity therefore was devised as an interdisciplinary platform for participants to discuss intracellular molecular transport from a variety of different perspectives. Topics covered at the meeting included RNA metabolism and transport, mechanisms of protein synthesis and localization, the formation of complex interactive protein ensembles, and the relevance of such mechanisms for activity-dependent regulation and synaptic plasticity in neurons. It was the overall objective of the colloquium to generate momentum and cohesion for the emerging research field of molecular kinesis.

Kinesin molecular motors: Transport pathways, receptors, and human disease

Kinesin molecular motor proteins are responsible for many of the major microtubule-dependent transport pathways in neuronal and non-neuronal cells. Elucidating the transport pathways mediated by kinesins, the identity of the cargoes moved, and the nature of the proteins that link kinesin motors to cargoes are areas of intense investigation. Kinesin-II recently was found to be required for transport in motile and nonmotile cilia and flagella where it is essential for proper left-right determination in mammalian development, sensory function in ciliated neurons, and opsin transport and viability in photoreceptors. Thus, these pathways and proteins may be prominent contributors to several human diseases including ciliary dyskinesias, situs inversus, and retinitis pigmentosa. Kinesin-I is needed to move many different types of cargoes in neuronal axons. Two candidates for receptor proteins that attach kinesin-I to vesicular cargoes were recently found. One candidate, sunday driver, is proposed to both link kinesin-I to an unknown vesicular cargo and to bind and organize the mitogen-activated protein kinase components of a c-Jun N-terminal kinase signaling module. A second candidate, amyloid precursor protein, is proposed to link kinesin-I to a different, also unknown, class of axonal vesicles. The finding of a possible functional interaction between kinesin-I and amyloid precursor protein may implicate kinesin-I based transport in the development of Alzheimer's disease.