A complex ligase ribozyme evolved in vitro from a group I ribozyme domain

Like most proteins, complex RNA molecules often are modular objects made up of distinct structural and functional domains. The component domains of a protein can associate in alternative combinations to form molecules with different functions. These observations raise the possibility that complex RNAs also can be assembled from preexisting structural and functional domains. To test this hypothesis, an in vitro evolution procedure was used to isolate a previously undescribed class of complex ligase ribozymes, starting from a pool of 1016 different RNA molecules that contained a constant region derived from a large structural domain that occurs within self-splicing group I ribozymes. Attached to this constant region were three hypervariable regions, totaling 85 nucleotides, that gave rise to the catalytic motif within the evolved catalysts. The ligase ribozymes catalyze formation of a 3′,5′-phosphodiester linkage between adjacent template-bound oligonucleotides, one bearing a 3′ hydroxyl and the other a 5′ triphosphate. Ligation occurs in the context of a Watson–Crick duplex, with a catalytic rate of 0.26 min−1 under optimal conditions. The constant region is essential for catalytic activity and appears to retain the tertiary structure of the group I ribozyme. This work demonstrates that complex RNA molecules, like their protein counterparts, can share common structural domains while exhibiting distinct catalytic functions.

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.

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.