Solution of the pulse width modulation problem using orthogonal polynomials and Korteweg-de Vries equations

The mathematical underpinning of the pulse width modulation (PWM) technique lies in the attempt to represent “accurately” harmonic waveforms using only square forms of a fixed height. The accuracy can be measured using many norms, but the quality of the approximation of the analog signal (a harmonic form) by a digital one (simple pulses of a fixed high voltage level) requires the elimination of high order harmonics in the error term. The most important practical problem is in “accurate” reproduction of sine-wave using the same number of pulses as the number of high harmonics eliminated. We describe in this paper a complete solution of the PWM problem using Padé approximations, orthogonal polynomials, and solitons. The main result of the paper is the characterization of discrete pulses answering the general PWM problem in terms of the manifold of all rational solutions to Korteweg-de Vries equations.

The negative feedback actions of progesterone on gonadotropinreleasing hormone secretion are transduced by the classical progesterone receptor

Progesterone (P) powerfully inhibits gonadotropin-releasing hormone (GnRH) secretion in ewes, as in other species, but the neural mechanisms underlying this effect remain poorly understood. Using an estrogen (E)-free ovine model, we investigated the immediate GnRH and luteinizing hormone (LH) response to acute manipulations of circulating P concentrations and whether this response was mediated by the nuclear P receptor. Simultaneous hypophyseal portal and jugular blood samples were collected over 36 hr: 0–12 hr, in the presence of exogenous P (P treatment begun 8 days earlier); 12–24 hr, P implant removed; 24–36 hr, P implant reinserted. P removal caused a significant rapid increase in the GnRH pulse frequency, which was detectable within two pulses (175 min). P insertion suppressed the GnRH pulse frequency even faster: the effect detectable within one pulse (49 min). LH pulsatility was modulated identically. The next two experiments demonstrated that these effects of P are mediated by the nuclear P receptor since intracerebroventricularly infused P suppressed LH release but 3α-hydroxy-5α-pregnan-20-one, which operates through the type A γ-aminobutyric acid receptor, was without effect and pretreatment with the P-receptor antagonist RU486 blocked the ability of P to inhibit LH. Our final study showed that P exerts its acute suppression of GnRH through an E-dependent system because the effects of P on LH secretion, lost after long-term E deprivation, are restored after 2 weeks of E treatment. Thus we demonstrate that P acutely inhibits GnRH through an E-dependent nuclear P-receptor system.

Double-level “orthogonal” dynamic combinatorial libraries on transition metal template

Dynamic combinatorial libraries are mixtures of compounds that exist in a dynamic equilibrium and can be driven to compositional self adaptation via selective binding of a specific assembly of certain components to a molecular target. We present here an extension of this initial concept to dynamic libraries that consists of two levels, the first formed by the coordination of terpyridine-based ligands to the transition metal template, and the second, by the imine formation with the aldehyde substituents on the terpyridine moieties. Dialdehyde 7 has been synthesized, converted into a variety of ligands, oxime ethers L11–L33 and acyl hydrazones L44–L77, and subsequently into corresponding cobalt complexes. A typical complex, Co(L22)22+ is shown to engage in rapid exchange with a competing ligand L11 and with another complex, Co(L22)22+ in 30% acetonitrile/water at pH 7.0 and 25°C. The exchange in the corresponding Co(III) complexes is shown to be much slower. Imine exchange in the acyl hydrazone complexes (L44–L77) is strongly controlled by pH and temperature. The two types of exchange, ligand and imine, can thus be used as independent equilibrium processes controlled by different types of external intervention, i.e., via oxidation/reduction of the metal template and/or change in the pH/temperature of the medium. The resulting double-level dynamic libraries are therefore named orthogonal, in similarity with the orthogonal protecting groups in organic synthesis. Sample libraries of this type have been synthesized and showed the complete expected set of components in electrospray ionization MS.

Orthogonal combinatorial mutagenesis: a codon-level combinatorial mutagenesis method useful for low multiplicity and amino acid-scanning protocols

We describe here a method to generate combinatorial libraries of oligonucleotides mutated at the codon-level, with control of the mutagenesis rate so as to create predictable binomial distributions of mutants. The method allows enrichment of the libraries with single, double or larger multiplicity of amino acid replacements by appropriate choice of the mutagenesis rate, depending on the concentration of synthetic precursors. The method makes use of two sets of deoxynucleoside-phosphoramidites bearing orthogonal protecting groups [4,4′-dimethoxytrityl (DMT) and 9-fluorenylmethoxycarbonyl (Fmoc)] in the 5′ hydroxyl. These phosphoramidites are divergently combined during automated synthesis in such a way that wild-type codons are assembled with commercial DMT-deoxynucleoside-methyl-phosphoramidites while mutant codons are assembled with Fmoc-deoxynucleoside-methyl-phosphoramidites in an NNG/C fashion in a single synthesis column. This method is easily automated and suitable for low mutagenesis rates and large windows, such as those required for directed evolution and alanine scanning. Through the assembly of three oligonucleotide libraries at different mutagenesis rates, followed by cloning at the polylinker region of plasmid pUC18 and sequencing of 129 clones, we concluded that the method performs essentially as intended.

A positivity result in the theory of Macdonald polynomials

We outline here a proof that a certain rational function Cn(q, t), which has come to be known as the “q, t-Catalan,” is in fact a polynomial with positive integer coefficients. This has been an open problem since 1994. Because Cn(q, t) evaluates to the Catalan number at t = q = 1, it has also been an open problem to find a pair of statistics a, b on the collection

Parametrizations of elliptic curves by Shimura curves and by classical modular curves

Fix an isogeny class

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.

Positive selection of an MHC class-I restricted TCR in the absence of classical MHC class I molecules

The H-2Ld alloreactive 2C T cell receptor (TCR) is commonly considered as being positively selected on the H-2Kb molecule. Surprisingly, 2C TCR+ CD8+ single-positive T cells emerge in massive numbers in fetal thymic organ culture originating from 2C transgenic, H-2KbDb−/− (2C+KbDb−/−) but not in fetal thymic organ culture from β2-microglobulin−/− 2C transgenic animals. Mature CD8+ T cells are observed in newborn but not in adult 2C+KbDb−/− mice. These CD8+ T cells express the α4β7 integrin, which allows them to populate the intestine, a pattern of migration visualized by intrathymic injection of FITC and subsequent accrual of FITC-labeled lymphocytes in the gut. We conclude that the 2C TCR is reactive not only with H-2Ld and H-2Kb, but also with nonclassical MHC class I products to enable positive selection of 2C+ T cells in the fetal and newborn thymus and to support their maintenance in the intestine.

Differential expression of the ToxR regulon in classical and E1 Tor biotypes of Vibrio cholerae is due to biotype-specific control over toxT expression.

The two major disease-causing biotypes of Vibrio cholerae, classical and El Tor, exhibit differences in their epidemic nature. Their behavior in the laboratory also differs in that El Tor strains produce two major virulence factors, cholera toxin (CT) and the toxin coregulated pilus (TCP), only under very restricted growth conditions, whereas classical strains do so in standard laboratory medium. Expression of toxin and TCP is controlled by two activator proteins, ToxR and ToxT, that operate in cascade fashion with ToxR controlling the synthesis of ToxT. Both biotypes express equivalent levels of ToxR, but only classical strains appear to express ToxT when grown in standard medium. In this report we show that restrictive expression of CT and TCP can be overcome in El Tor strains by expressing ToxT independently of ToxR. An El Tor strain lacking functional ToxT does not express CT or TCP, ruling out existence of a cryptic pathway for virulence regulation in this biotype. These results may have implications for understanding the evolution of El Tor strains toward reduced virulence with respect to classical strains.

Mutation of a conserved serine in TM4 of opioid receptors confers full agonistic properties to classical antagonists.

The involvement of a conserved serine (Ser196 at the mu-, Ser177 at the delta-, and Ser187 at the kappa-opioid receptor) in receptor activation is demonstrated by site-directed mutagenesis. It was initially observed during our functional screening of a mu/delta-opioid chimeric receptor, mu delta2, that classical opioid antagonists such as naloxone, naltrexone, naltriben, and H-Tyr-Tic[psi,CH2NH]Phe-Phe-OH (TIPPpsi; Tic = 1,2,3,4-tetrahydroisoquinoline-3-carboxylic acid) could inhibit forskolin-stimulated adenylyl cyclase activity in CHO cells stably expressing the chimeric receptor. Antagonists also activated the G protein-coupled inward rectifying potassium channel (GIRK1) in Xenopus oocytes coexpressing the mu delta2 opioid receptor and the GIRK1 channel. By sequence analysis and back mutation, it was determined that the observed antagonist activity was due to the mutation of a conserved serine to leucine in the fourth transmembrane domain (S196L). The importance of this serine was further demonstrated by analogous mutations created in the mu-opioid receptor (MORS196L) and delta-opioid receptor (DORS177L), in which classical opioid antagonists could inhibit forskolin-stimulated adenylyl cyclase activity in CHO cells stably expressing either MORS196L or DORS177L. Again, antagonists could activate the GIRK1 channel coexpressed with either MORS196L or DORS177L in Xenopus oocytes. These data taken together suggest a crucial role for this serine residue in opioid receptor activation.

Local densities orthogonal to beta-sheet amide planes: patterns of packing in globular proteins.

We have investigated the efficiency of packing by calculating intramolecular packing density above and below peptide planes of internal beta-pleated sheet residues in five globular proteins. The orientation of interest was chosen to allow study of regions that are approximately perpendicular to the faces of beta-pleated sheets. In these locations, nonbonded van der Waals packing interactions predominate over hydrogen bonding and solvent interactions. We observed considerable variability in packing densities within these regions, confirming that the interior packing of a protein does not result in uniform occupation of the available space. Patterns of fluctuation in packing density suggest that the regular backbone-to-backbone network of hydrogen bonds is not likely to be interrupted to maximize van der Waals interactions. However, high-density packing tends to occur toward the ends of beta-structure strands where hydrogen bonds are more likely to involve nonpolar side-chain groups or solvent molecules. These features result in internal protein folding with a central low-density core surrounded by a higher-density subsurface shell, consistent with our previous calculations regarding overall protein packing density.

Peptide synthesis using unprotected peptides through orthogonal coupling methods.

We describe an approach to the synthesis of peptides from segments bearing no protecting groups through an orthogonal coupling method to capture the acyl segment as a thioester that then undergoes an intramolecular acyl transfer to the amine component with formation of a peptide bond. Two orthogonal coupling methods to give the covalent ester intermediate were achieved by either a thiol-thioester exchange mediated by a trialkylphosphine and an alkylthiol or a thioesterification by C alpha-thiocarboxylic acid reacting with a beta-bromo amino acid. With this approach, unprotected segments ranging from 4 to 37 residues were coupled to aqueous solution to give free peptides up to 54 residues long with high efficiency.