Prime Residues Research Articles

On The Ono Invariants of Imaginary Quadratic Fields

Let Ed(x) denote the “Euler polynomial” x2+x+(1−d)/4 if d≡1 (mod 4) and x2−d if d≡2,3 (mod 4). Set Ω(n)=the number of prime factors (counting multiplicity) of the positive integer n. The Ono invariantOnod of K is defined to be max{Ω(Ed(b)): b=0,1,…∣Δd∣/4−1∣ except when d=−1,−3 in which case Onod is defined to be 1. Finally, let hd=hK denote the class number of K=Q(d). It is known that hd=1⇔Onod=1 for all negative integers d (Frobenius–Rabinowitch). We improve a result of Chowla–Cowles–Cowles and characterize imaginary quadratic fields of class number one in terms of least prime quadratic residues. This yields an elementary characterization of imaginary quadratic fields with Ono invariant equal to one. It is also known that hd=2⇔Onod=2 for all negative integers d (Sasaki). Sasaki also proved that hd⩾Onod for all negative integers d. If hd=3, then necessarily Onod=3 and −d is a prime p≡3 (mod 4). Computer calculations support the conjecture that the converse holds. We prove in this paper that this conjecture fails for at most finitely many d. We will show in fact that there are only finitely many d with −d a prime p≡3 (mod 4) and Onod=3. Moreover, using a result of Bach (which assumes the extended Riemann hypothesis), we find that the conjecture holds for all −d=prime p≡3 (mod 4) greater than 1017. Computer calculations so far show that the conjecture holds up to 1.5×107. Another conditional result, derived from Lagarias–Odlyzko (Effective versions of the Chebotarev density theorem, in: “Algebraic Number Fields” (A. Frohlich, Ed.), Academic Press, London, 1977), which assumes GRH, is limd→−∞Onod=∞.

Activity of Recombinant Dengue 2 Virus NS3 Protease in the Presence of a Truncated NS2B Co-factor, Small Peptide Substrates, and Inhibitors

Recombinant forms of the dengue 2 virus NS3 protease linked to a 40-residue co-factor, corresponding to part of NS2B, have been expressed in Escherichia coli and shown to be active against para-nitroanilide substrates comprising the P6-P1 residues of four substrate cleavage sequences. The enzyme is inactive alone or after the addition of a putative 13-residue co-factor peptide but is active when fused to the 40-residue co-factor, by either a cleavable or a noncleavable glycine linker. The NS4B/NS5 cleavage site was processed most readily, with optimal processing conditions being pH 9, I = 10 mm, 1 mm CHAPS, 20% glycerol. A longer 10-residue peptide corresponding to the NS2B/NS3 cleavage site (P6-P4') was a poorer substrate than the hexapeptide (P6-P1) para-nitroanilide substrate under these conditions, suggesting that the prime side substrate residues did not contribute significantly to protease binding. We also report the first inhibitors of a co-factor-complexed, catalytically active flavivirus NS3 protease. Aprotinin was the only standard serine protease inhibitor to be active, whereas a number of peptide substrate analogues were found to be competitive inhibitors at micromolar concentrations.

Expression and Purification of Recombinant Rhinovirus 14 3CD Proteinase and Its Comparison to the 3C Proteinase

Human rhinovirus (HRV) is a positive-stranded RNA virus with an open reading frame that encodes for a single polyprotein of about 3000 amino acids. The HRV polyprotein is proteolytically processed; eight of nine cleavages are catalyzed by the 3C and/or the 3CD proteinases. We have expressed and purified recombinant HRV14 3C and 3CD proteinases and investigated their substrate selectivity and inhibitor sensitivity. Expressed 3CD proteinase had the P1/P1′ residues of the 3C/3D cleavage site mutated from Gln/Gly to Ala/Ala in order to prevent autocleavage. The 3CD proteinase activities were measured by utilization of native, chromogenic, and fluorogenic peptide substrates. The 3CD proteinase exhibited ≤15% activity, compared to 3C, toward peptidylp-nitroanilide substrates which contain only thep-nitroaniline moiety in the prime side. The 3C and 3CD proteinases exhibited similar activities for both internally quenched fluorogenic and native peptides. These results suggest that the two enzymes have similar but slightly different substrate specificity, especially on their preference for prime side residues. Inhibitor sensitivities toward classical proteinase inhibitors were generally similar for both enzymes. Small peptidyl inhibitors, specifically designed and synthesized for HRV14 3C, also inhibited the 3CD proteinase. Taken together, our data indicate that the 3D domain of 3CD proteinase had some influence on substrate recognition, but did not have dramatic impact on its interaction with inhibitors.

Mechanistic role of an NS4A peptide cofactor with the truncated NS3 protease of hepatitis C virus: elucidation of the NS4A stimulatory effect via kinetic analysis and inhibitor mapping.

Infection by hepatitis C viruses (HCVs) is a serious medical problem with no broadly effective treatment available for the progression of chronic hepatitis. The catalytic activity of a viral serine protease located in the N-terminal one-third of nonstructural protein 3 (NS3) is required for polyprotein processing at four site-specific junctions. The three-dimensional crystal structure of the NS3-NS4A co-complex [Kim, J. L., Morgenstern, K. A., Lin, C., Fox, T., Dwyer, M. D., Landro, J. A., Chambers, S. P., Markland, W., Lepre, C. A., O'Malley, E. T., Harbeson, S. L., Rice, C. M., Murcko, M. A., Caron, P. R., & Thomson, J. A. (1996) Cell 87, 343-355] delineates a small hydrophobic region within the 54-residue NS4A protein that intercalates with and makes extensive contacts to the core of the protease. The current investigation addresses the mechanism of NS3 protease catalytic activation by NS4A utilizing a small synthetic NS4A peptide (residues 1678-1691 of the virus polyprotein sequence) and the recombinantly expressed protease domain of NS3. The addition of NS4A dramatically increased NS3 kcat and kcat/Km catalytic parameters when measured against small peptide substrates representing the different site-specific junctions of the polyprotein. The catalytic effect of natural and non-natural amino acid substitutions at the P1 position in a 5A/5B peptide substrate was investigated. NS3-NS4A demonstrated a marked catalytic preference for the cysteine residue commonly found in authentic substrates. The pH dependence of the NS3 hydrolysis reaction is not affected by the presence of NS4A. This result suggests that NS4A does not change the pKa values of the active site residues of NS3 protease. A steady state kinetic analysis was performed and indicated that the binding of NS4A and the peptide substrate occurs in an ordered fashion during the catalytic cycle, with NS4A binding first. Two distinct kinetic classes of peptidyl inhibitors based upon the 5A/5B cleavage site were identified. An NS4A-independent class is devoid of prime residues. A second class of inhibitors is NS4A-dependent and contains a natural or non-natural cyclic amino acid substituted for the commonly found P1' residue serine. These inhibitors display an up to 80-fold increase in affinity for NS3 protease in the presence of NS4A. Sequential truncation of prime and P residues from this inhibitor class demonstrated the fact that the P4' and P1' residues are crucial for potent inhibition. The selectivity of this NS4A effect is interpreted using a model of the 5A/5B decapeptide substrate bound to the active site of the NS3-NS4A structure.

Kininogen-derived fluorogenic substrates for investigating the vasoactive properties of rat tissue kallikreins--identification of a T-kinin-releasing rat kallikrein.

Peptide substrates with intramolecularly quenched fluorescence that reproduce the rat kininogen sequences at both ends of the bradykinin moiety were synthesized and used to investigate the kinin-releasing properties of five rat tissue kallikreins (rK1, rK2, rK7, rK9, rK10). Substrates derived from rat H- and L-kininogen were cleaved best by rK1, especially that including the N-terminal insertion site of bradykinin, Abz-TSVIRRPQ-EDDnp(Abz = O-aminobenzoyl, EDDnp = ethylenediamine 2,4-dinitrophenyl), which was cleaved at the R-R bond with a k(cat)/Km of 12400 mM(-1) s(-1). Replacement of the P2' residue Pro by Val in Abz-TSVIRRPQ-EDDnp gave a far less specific substrate that was rapidly hydrolysed by all five rat kallikreins and human kallikrein hK1. Peptidyl-N-methyl coumarylamide substrates, which lack prime residues, also had low specificities. The importance of the P2' residue for rK1 specificity was further demonstrated using a human-kininogen-derived substrate that included the N-terminal insertion site of bradykinin (Abz-LMKRP-EDDnp). This was cleaved at the M-K bond by hK1 (kallidin-releasing site), but at the K-R bond (bradykinin-releasing site) by rK1. Competition experiments with Abz-TSVIRRPQ-EDDnp, which is resistant to most kallikreins, and Abz-TSVIRRVQ-EDDnp, a general kallikrein substrate, demonstrated that the former competitively inhibited hydrolysis by rK9 and hK1, with Ki values similar to the Km values for the substrate. Thus Pro in P2' does not prevent the peptide binding to the enzyme active site, but impairs cleavage of the scissile bond. The T-kininogen-derived substrate with the T-kinin C-terminal sequence (Abz-FRLVR-EDDnp) was cleaved by rK10 (k(cat)/Km = 2310 mM(-1) s(-1)) and less rapidly by rK1, rK7 and hK1, at the R-L bond, while that corresponding to the N-terminal (Abz-ALDMMISRP-EDDnp) of T-kinin was resistant to all five kallikreins used, suggesting that none has T-kininogenase activity. But this substrate was hydrolysed by a semipurified sample of submandibular gland extract. Another kallikrein, identified as kallikrein rK3, was isolated from this fraction and shown to hydrolyze Abz-ALDMMISRP-EDDnp; rK3 also specifically released T-kinin from purified T1/T2-kininogen after HPLC fractionation. Injection of purified rK3 and of Abz-ALDMMISRP-EDDnp-cleaving fractions into the circulation of anesthesized rats caused transient falls in blood pressure, as did purified rK1 but none of the other purified rat or human kallikreins. This effect occurred via activation of the kinin system since it was blocked by Hoe140, a kinin receptor antagonist.

On abelian difference set codes

In this paper we determine the ranks of the incidence matrices that belong to the following types of difference sets: Twin prime power difference sets, biquadratic residues and biquadratic residues with 0. We also prove a conjecture of Assmus and Key on the code generated by the hyperovals of PG(2, q).

Gas chromatographic determination of flumetralin in tobacco

A method for the determination of flumetralin (Prime) residues in tobacco is presented. The procedure is adapted from that of Reif and Moser, which was proposed for the determination of organochlorine pesticide residues in tobacco. Improvements were made by the use of an overflow in the Soxhlet extractor and by replacing the packed column with a fused-silica capillary column. The suggested method is rapid and simple to use.

Sequences in power residue classes

Using A. Well′s estimates the authors have given bounds for the largest prime P0 such that all primes p > P0 have sequences of quadratic residues of length m. For m ≤ 8 the smallest prime having m consecutive quadratic residues is ≡3(mod4) and P0 ≡ 1(mod4). The reason for this phenomenon is investigated in this paper and the theory developed is used to successfully uncover analogous phenomena for rth power residues, r ≥ 2, r even.

Cyclic properties of pseudo-random sequences of Mersenne prime residues

In Lehmer's multiplicative congruential procedure for generating a sequence of pseudo-random numbers, the modulus may be chosen as a Mersenne prime of the form, Mp = 2p − 1, and one of its primitive roots used as the constant multiplier to ensure a maximal sequence. Cyclic properties of the sequence entail perfect negative correlation between halves of the sequence and other relationships which limit the useful sequence length. A primitive root is shown to be characterised by a set of non-trivial roots of unity (mod Mp), which is used to identify a primitive root, and properties of finite rings of such roots are used to generate further primitive roots. Computer procedures to facilitate these operations are indicated and applied to production of pseudo-random n-tuples designed to overcome the restrictions on randomness of single generator n-tuples, notes in the literature.