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.

Heteroduplex joint formation in Escherichia coli recombination is initiated by pairing of a 3′-ending strand

The formation of heteroduplex joints in Escherichia coli recombination is initiated by invasion of double-stranded DNA by a single-stranded homologue. To determine the polarity of the invasive strand, linear molecules with direct terminal repeats were released by in vivo restriction of infecting chimeric phage DNA and heteroduplex products of intramolecular recombination were analyzed. With this substrate, the invasive strand is expected to be incorporated into the circular crossover product and the complementary strand is expected to be incorporated into the reciprocal linear product. Strands of both polarities were incorporated into heteroduplex structures, but only strands ending 3′ at the break were incorporated into circular products. This result indicates that invasion of the 3′-ending strand initiates the heteroduplex joint formation and that the complementary 5′-ending strand is incorporated into heteroduplex structures in the process of reciprocal strand exchange. The polarity of the invasive strand was not affected by recD, recJ, or xonA mutations. However, xonA and recJ mutations increased the proportion of heteroduplexes containing 5′-ending strands. This observation suggests that RecJ exonuclease and exonuclease I may enhance recombination by degrading the displaced strands during branch migration and thereby causing strand exchange to be unidirectional.

Content mixing and membrane integrity during membrane fusion driven by pairing of isolated v-SNAREs and t-SNAREs

Membrane bilayer fusion has been shown to be mediated by v- and t-SNAREs initially present in separate populations of liposomes and to occur with high efficiency at a physiologically meaningful rate. Lipid mixing was demonstrated to involve both the inner and the outer leaflets of the membrane bilayer. Here, we use a fusion assay that relies on duplex formation of oligonucleotides introduced in separate liposome populations and report that SNARE proteins suffice to mediate complete membrane fusion accompanied by mixing of luminal content. We also find that SNARE-mediated membrane fusion does not compromise the integrity of liposomes.

A thymidine triphosphate shape analog lacking Watson–Crick pairing ability is replicated with high sequence selectivity

Compound 1 (F), a nonpolar nucleoside analog that is isosteric with thymidine, has been proposed as a probe for the importance of hydrogen bonds in biological systems. Consistent with its lack of strong H-bond donors or acceptors, F is shown here by thermal denaturation studies to pair very poorly and with no significant selectivity among natural bases in DNA oligonucleotides. We report the synthesis of the 5′-triphosphate derivative of 1 and the study of its ability to be inserted into replicating DNA strands by the Klenow fragment (KF, exo− mutant) of Escherichia coli DNA polymerase I. We find that this nucleotide derivative (dFTP) is a surprisingly good substrate for KF; steady-state measurements indicate it is inserted into a template opposite adenine with efficiency (Vmax/Km) only 40-fold lower than dTTP. Moreover, it is inserted opposite A (relative to C, G, or T) with selectivity nearly as high as that observed for dTTP. Elongation of the strand past F in an F–A pair is associated with a brief pause, whereas that beyond A in the inverted A–F pair is not. Combined with data from studies with F in the template strand, the results show that KF can efficiently replicate a base pair (A–F/F–A) that is inherently very unstable, and the replication occurs with very high fidelity despite a lack of inherent base-pairing selectivity. The results suggest that hydrogen bonds may be less important in the fidelity of replication than commonly believed and that nucleotide/template shape complementarity may play a more important role than previously believed.

RecA tests homology at both pairing and strand exchange

RecA is a 38-kDa protein from Escherichia coli that polymerizes on single-stranded DNA, forming a nucleoprotein filament that pairs with homologous duplex DNA and carries out strand exchange in vitro. To observe the effects of mismatches on the kinetics of the RecA-catalyzed recombination reaction, we used assays based upon fluorescence energy transfer that can differentiate between the pairing and strand displacement phases. Oligonucleotide sequences that produced 2–14% mismatches in the heteroduplex product of strand exchange were tested, as well as completely homologous and heterologous sequences. The equilibrium constant for pairing decreased as the number of mismatches increased, which appeared to result from both a decrease in the rate of formation and an increase in the rate of dissociation of the intermediates. In addition, the rate of strand displacement decreased with increasing numbers of mismatches, roughly in proportion to the number of mismatches. The equilibrium constant for pairing and the rate constant for strand displacement both decreased 6-fold as the heterology increased to 14%. These results suggest that discrimination of homology from heterology occurs during both pairing and strand exchange.

A curved RNA helix incorporating an internal loop with G·A and A·A non-Watson–Crick base pairing

The crystal structure of the RNA dodecamer 5′-GGCC(GAAA)GGCC-3′ has been determined from x-ray diffraction data to 2.3-Å resolution. In the crystal, these oligomers form double helices around twofold symmetry axes. Four consecutive non-Watson–Crick base pairs make up an internal loop in the middle of the duplex, including sheared G·A pairs and novel asymmetric A·A pairs. This internal loop sequence produces a significant curvature and narrowing of the double helix. The helix is curved by 34° from end to end and the diameter is narrowed by 24% in the internal loop. A Mn2+ ion is bound directly to the N7 of the first guanine in the Watson–Crick region following the internal loop and the phosphate of the preceding residue. This Mn2+ location corresponds to a metal binding site observed in the hammerhead catalytic RNA.

The crystal structure of the Rev binding element of HIV-1 reveals novel base pairing and conformational variability

The crystal and molecular structure of an RNA duplex corresponding to the high affinity Rev protein binding element (RBE) has been determined at 2.1-Å resolution. Four unique duplexes are present in the crystal, comprising two structural variants. In each duplex, the RNA double helix consists of an annealed 12-mer and 14-mer that form an asymmetric internal loop consisting of G-G and G-A noncanonical base pairs and a flipped-out uridine. The 12-mer strand has an A-form conformation, whereas the 14-mer strand is distorted to accommodate the bulges and noncanonical base pairing. In contrast to the NMR model of the unbound RBE, an asymmetric G-G pair with N2-N7 and N1-O6 hydrogen bonding, is formed in each helix. The G-A base pairing agrees with the NMR structure in one structural variant, but forms a novel water-mediated pair in the other. A backbone flip and reorientation of the G-G base pair is required to assume the RBE conformation present in the NMR model of the complex between the RBE and the Rev peptide.

Computation, prediction, and experimental tests of fitness for bacteriophage T7 mutants with permuted genomes

We created a simulation based on experimental data from bacteriophage T7 that computes the developmental cycle of the wild-type phage and also of mutants that have an altered genome order. We used the simulation to compute the fitness of more than 105 mutants. We tested these computations by constructing and experimentally characterizing T7 mutants in which we repositioned gene 1, coding for T7 RNA polymerase. Computed protein synthesis rates for ectopic gene 1 strains were in moderate agreement with observed rates. Computed phage-doubling rates were close to observations for two of four strains, but significantly overestimated those of the other two. Computations indicate that the genome organization of wild-type T7 is nearly optimal for growth: only 2.8% of random genome permutations were computed to grow faster, the highest 31% faster, than wild type. Specific discrepancies between computations and observations suggest that a better understanding of the translation efficiency of individual mRNAs and the functions of qualitatively “nonessential” genes will be needed to improve the T7 simulation. In silico representations of biological systems can serve to assess and advance our understanding of the underlying biology. Iteration between computation, prediction, and observation should increase the rate at which biological hypotheses are formulated and tested.

New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function

In this paper, we give two infinite families of explicit exact formulas that generalize Jacobi’s (1829) 4 and 8 squares identities to 4n2 or 4n(n + 1) squares, respectively, without using cusp forms. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results arise in the setting of Jacobi elliptic functions, Jacobi continued fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. We have also obtained many additional infinite families of identities in this same setting that are analogous to the η-function identities in appendix I of Macdonald’s work [Macdonald, I. G. (1972) Invent. Math. 15, 91–143]. A special case of our methods yields a proof of the two conjectured [Kac, V. G. and Wakimoto, M. (1994) in Progress in Mathematics, eds. Brylinski, J.-L., Brylinski, R., Guillemin, V. & Kac, V. (Birkhäuser Boston, Boston, MA), Vol. 123, pp. 415–456] identities involving representing a positive integer by sums of 4n2 or 4n(n + 1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson’s Cℓ nonterminating 6φ5 summation theorem, and Andrews’ basic hypergeometric series proof of Jacobi’s 4 and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n2 or n(n + 1) squares, respectively. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Kac and Wakimoto, and many others.

Molecular computation: RNA solutions to chess problems

We have expanded the field of “DNA computers” to RNA and present a general approach for the solution of satisfiability problems. As an example, we consider a variant of the “Knight problem,” which asks generally what configurations of knights can one place on an n × n chess board such that no knight is attacking any other knight on the board. Using specific ribonuclease digestion to manipulate strands of a 10-bit binary RNA library, we developed a molecular algorithm and applied it to a 3 × 3 chessboard as a 9-bit instance of this problem. Here, the nine spaces on the board correspond to nine “bits” or placeholders in a combinatorial RNA library. We recovered a set of “winning” molecules that describe solutions to this problem.

Twist fields, the elliptic genus, and hidden symmetry

We combine infinite dimensional analysis (in particular a priori estimates and twist positivity) with classical geometric structures, supersymmetry, and noncommutative geometry. We establish the existence of a family of examples of two-dimensional, twist quantum fields. We evaluate the elliptic genus in these examples. We demonstrate a hidden SL(2,ℤ) symmetry of the elliptic genus, as suggested by Witten.

Homologous pairing in stretched supercoiled DNA

By using elastic measurements on single DNA molecules, we show that stretching a negatively supercoiled DNA activates homologous pairing in physiological conditions. These experiments indicate that a stretched unwound DNA locally denatures to alleviate the force-driven increase in torsional stress. This is detected by hybridization with 1 kb of homologous single-stranded DNA probes. The stretching force involved (≈2 pN) is small compared with those typically developed by molecular motors, suggesting that this process may be relevant to DNA processing in vivo. We used this technique to monitor the progressive denaturation of DNA as it is unwound and found that distinct, stable denaturation bubbles formed, beginning in A+T-rich regions.

Homologous-pairing activity of the human DNA-repair proteins Xrcc3⋅Rad51C

The human Xrcc3 protein is involved in the repair of damaged DNA through homologous recombination, in which homologous pairing is a key step. The Rad51 protein is believed to be the only protein factor that promotes homologous pairing in recombinational DNA repair in mitotic cells. In the brain, however, Rad51 expression is extremely low, whereas XRCC3, a human homologue of Saccharomyces cerevisiae RAD57 that activates the Rad51-dependent homologous pairing with the yeast Rad55 protein, is expressed. In this study, a two-hybrid analysis conducted with the use of a human brain cDNA library revealed that the major Xrcc3-interacting protein is a Rad51 paralog, Rad51C/Rad51L2. The purified Xrcc3⋅Rad51C complex, which shows apparent 1:1 stoichiometry, was found to catalyze the homologous pairing. Although the activity is reduced, the Rad51C protein alone also catalyzed homologous pairing, suggesting that Rad51C is a catalytic subunit for homologous pairing. The DNA-binding activity of Xrcc3⋅Rad51C was drastically decreased in the absence of Xrcc3, indicating that Xrcc3 is important for the DNA binding of Xrcc3⋅Rad51C. Electron microscopic observations revealed that Xrcc3⋅Rad51C and Rad51C formed similar filamentous structures with circular single-stranded DNA.