Arithmetic of cyclotomic fields and Fermat-type equations

Let l be any odd prime, and ζ a primitive l-th root of unity. Let C_l be the l-Sylow subgroup of the ideal class group of Q(ζ). The Teichmüller character w : Z_l → Z^*_l is given by w(x) = x (mod l), where w(x) is a p-1-st root of unity, and x ∈ Z_l. Under the action of this character, C_l decomposes as a direct sum of C^((i))_l, where C^((i))_l is the eigenspace corresponding to w^i. Let the order of C^((3))_l be l^h_3). The main result of this thesis is the following: For every n ≥ max( 1, h_3 ), the equation x^(ln) + y^(ln) + z^(ln) = 0 has no integral solutions (x,y,z) with l ≠ xyz. The same result is also proven with n ≥ max(1,h_5), under the assumption that C_l^((5)) is a cyclic group of order l^h_5. Applications of the methods used to prove the above results to the second case of Fermat's last theorem and to a Fermat-like equation in four variables are given.

The proof uses a series of ideas of H.S. Vandiver ([Vl],[V2]) along with a theorem of M. Kurihara [Ku] and some consequences of the proof of lwasawa's main conjecture for cyclotomic fields by B. Mazur and A. Wiles [MW]. In [V1] Vandiver claimed that the first case of Fermat's Last Theorem held for l if l did not divide the class number h^+ of the maximal real subfield of Q(e^(2πi/i)). The crucial gap in Vandiver's attempted proof that has been known to experts is explained, and complete proofs of all the results used from his papers are given.

Studies of the serotonin type 3A receptor and the chemical preparation of tRNA

This thesis describes studies surrounding a ligand-gated ion channel (LGIC): the serotonin type 3A receptor (5-HTb>3b>AR). Structure-function experiments using unnatural amino acid mutagenesis are described, as well as experiments on the methodology of unnatural amino acid mutagenesis. Chapter 1 introduces LGICs, experimental methods, and an overview of the unnatural amino acid mutagenesis.

In Chapter 2, the binding orientation of the clinically available drugs ondansetron and granisetron within 5-HTb>3b>A is determined through a combination of unnatural amino acid mutagenesis and an inhibition based assay. A cation-π interaction is found for both ondansetron and granisetron with a specific tryptophan residue (Trp183, TrpB) of the mouse 5-HTb>3b>AR, which establishes a binding orientation for these drugs.

In Chapter 3, further studies were performed with ondansetron and granisetron with 5-HTb>3b>A. The primary determinant of binding for these drugs was determined to not include interactions with a specific tyrosine residue (Tyr234, TyrC2). In completing these studies, evidence supporting a cation-π interaction of a synthetic agonist, meta-chlorophenylbiguanide, was found with TyrC2.

In Chapter 4, a direct chemical acylation strategy was implemented to prepare full-length suppressor tRNA mediated by lanthanum(III) and amino acid phosphate esters. The derived aminoacyl-tRNA is shown to be translationally competent in Xenopus oocytes.

Appendix A.1 gives details of a pharmacological method for determining the equilibrium dissociation constant, Kb>Bb>, of a competitive antagonist with a receptor, known as Schild analysis. Appendix A.2 describes an examination of the inhibitory activity of new chemical analogs of the 5-HTb>3b>A antagonist ondansetron. Appendix A.3 reports an organic synthesis of an intermediate for a new unnatural amino acid. Appendix A.4 covers an additional methodological examination for the preparation of amino-acyl tRNA.

Binding Site Structure and Stoichiometry in Serotonin Type 3 Receptors

This dissertation primarily describes studies of serotonin type 3 (5-HTb>3b>) receptors of the Cys-loop super-family of ligand gated ion channels. The first chapter provides a general introduction to these important proteins and the methods used to interrogate their structure and function. The second chapter details the delineation of a structural unit of the ligand binding site of homomeric 5-HTb>3b>A receptors on which the ligands serotonin (5-HT) and m-chlorophenyl biguanide (mCPBG) are reliant for effective receptor activation. Unnatural amino acid mutagenesis results show that the details of each ligand’s interaction with this organizing feature of the binding site differ, providing insights into general principles of receptor activation.

The third chapter describes a study in which florescent protein fusions of the A and B subunits of the heteromeric 5-HTb>3b>AB receptor are employed to determine the subunit stoichiometry and order within functional receptors. Strong evidence is found for an Ab>3b>Bb>2b> stoichiometry with A-A-B-A-B order. The fourth chapter investigates the potential for ligand binding across heteromeric binding sites in the 5-HTb>3b>AB receptor. Unlike serotonin, mCPBG is found to bind the receptor at heteromeric binding sites. Further mCPBG is capable of allosterically modulating the response of serotonin on the 5-HTb>3b>AB receptor from these heteromeric sites.

Finally, the fifth chapter describes progress towards the application of unnatural amino acid mutagenesis to an important new class of proteins, transcription factors. Experiments optimizing novel methods for the detection of function are described, using RARα of the nuclear receptor family of transcription factors.

Part I. Coenzyme B12 as a hydroformylation-type catalyst. Part II. Mechanisms of hydrogen transfer in the methylmalonyl coenzyme a mutase reaction

Part I

The mechanism of the hydroformylation reaction was studied. Using cobalt deuterotetracarbonyl and 1-pentene as substrates, the first step in the reaction, addition of cobalt tetracarbonyl to an olefin, was shown to be reversible.

Part II

The role of coenzyme Bb>12b> in the isomerization of methylmalonyl coenzyme A to succinyl coenzyme A by methylmalonyl coenzyme A mutase was studied. The reaction was allowed to proceed to partial completion using a mixture of methylmalonyl coenzyme A and 4, 4, 4-tri-²H-methylmalonyl coenzyme A as substrate. The deuterium distribution in the product, succinyl coenzyme A, was shown to best fit a model in which hydrogen is transferred from C-4 of methylmalonyl coenzyme A to C-5’ of the adenosyl moiety of coenzyme Bb>12b> in the rate determining step. The three hydrogens at the 5’-adenosyl position of the coenzyme Bb>12b> intermediate are then able to become enzymatically equivalent before hydrogen is transferred from the coenzyme Bb>12b> intermediate to form succinyl coenzyme A.