Constructing cocyclic Hadamard matrices of order 4p

Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices (HMs) with interesting algebraic properties. Ó Catháin and Röder described a classification algorithm for CHMs of order $4n$ based on relative difference sets in groups of order $8n$; this led to the classification of all CHMs of order at most 36. On the basis of work of de Launey and Flannery, we describe a classification algorithm for CHMs of order $4p$ with $p$ a prime; we prove refined structure results and provide a classification for $p\le 13$. Our analysis shows that every CHM of order $4p$ with $p\equiv 1mod4$ is equivalent to a HM with one of five distinct block structures, including Williamson‐type and (transposed) Ito matrices. If $p\equiv 3mod4$, then every CHM of order $4p$ is equivalent to a Williamson‐type or (transposed) Ito matrix.     

Relative Difference Sets and Hadamard Matrices from Perfect Quaternionic Arrays

Let \(G=\mathbf{C}_{n_1}\times \cdots \times \mathbf{C}_{n_m}\) be an abelian group of order \(n=n_1\dots n_m\), where each \(\mathbf{C}_{n_t}\) is cyclic of order \(n_t\). We present a correspondence between the (4n, 2, 4n, 2n)-relative difference sets in \(G\times Q_8\) relative to the centre \(Z(Q_8)\) and the perfect arrays of size \(n_1\times \dots \times n_m\) over the quaternionic alphabet \(Q_8\cup qQ_8\), where \(q=(1+i+j+k)/2\). In view of this connection, for \(m=2\) we introduce new families of relative difference sets in \(G\times Q_8\), as well as new families of Williamson and Ito Hadamard matrices with G-invariant components.     

Inter- and intramolecular mechanisms for chlorine rearrangements in trimethyl-substituted N-chlorohydantoins

The antimicrobial compounds 1-chloro-3,5,5-trimethylhydantoin and 3-chloro-1,5,5-trimethylhydantoin (1 and 2, respectively) have been synthesized and examined via a joint experimental and computational study. The measured rate of loss of oxidative chlorine in the absence and presence of exposure to UVA irradiation determined 2 to be less stable than 1. An interesting migration reaction was observed during UVA irradiation that featured the production of chlorine rearrangement and dechlorinated compounds. Two novel hydrogen atom transfer reaction (HATR) mechanisms have been proposed: (1) an intramolecular process in which a hydrogen atom undergoes a series of sigmatropic shifts, and (2) an intermolecular pathway in which a radical abstracts a hydrogen atom from a neighboring molecule. Density functional theory (DFT) calculations at the UB3LYP/6-311G++(2d,p) theory level have been employed to elucidate the preferred reaction pathway. Both proposed HATR mechanisms predicted 2 to possess a lower free energy of activation, ΔG(?), relative to 1 in accordance with the experimental stability measurements. However, the intermolecular route had an overall lower absolute ΔG(?) and was more consistent with measured product ratios in solution. The intermolecular reaction pathway, unlike the intramolecular route, also predicted the lack of formation of a migration product featuring a Cl covalently bonded to a methylene group at the 5-position of the hydantoin moiety, which was verified by NMR experiments.     

Regio‐ and Stereoselective Steroid Hydroxylation at C7 by Cytochrome P450 Monooxygenase Mutants

Steroidal C7β alcohols and their respective esters have shown significant promise as neuroprotective and anti‐inflammatory agents to treat chronic neuronal damage like stroke, brain trauma, and cerebral ischemia. Since C7 is spatially far away from any functional groups that could direct C?H activation, these transformations are not readily accessible using modern synthetic organic techniques. Reported here are P450‐BM3 mutants that catalyze the oxidative hydroxylation of six different steroids with pronounced C7 regioselectivities and β stereoselectivities, as well as high activities. These challenging transformations were achieved by a focused mutagenesis strategy and application of a novel technology for protein library construction based on DNA assembly and USER (Uracil‐Specific Excision Reagent) cloning. Upscaling reactions enabled the purification of the respective steroidal alcohols in moderate to excellent yields. The high‐resolution X‐ray structure and molecular dynamics simulations of the best mutant unveil the origin of regio‐ and stereoselectivity.     

Computational insight into small molecule inhibition of cyclophilins

Cyclophilins (Cyp) are a family of cellular enzymes possessing peptidyl-prolyl isomerase activity, which catalyze the cis-trans interconversion of proline-containing peptide bonds. The two most abundant family members, CypA and CypB, have been identified as valid drug targets for a wide range of diseases, including HCV, HIV, and multiple cancers. However, the development of small molecule inhibitors that possess nM potency and high specificity for a particular Cyp is difficult given the complete conservation of all active site residues between the enzymes. Monte Carlo statistical sampling coupled to free energy perturbation theory (MC/FEP) calculations have been carried out to elucidate the origin of the experimentally observed nM inhibition of CypA by acylurea-based derivatives and the >200-fold in vitro selectivity between CypA and CypB from aryl 1-indanylketone-based μM inhibitors. The computed free-energies of binding were in close accord with those derived from experiments. Binding affinity values for the inhibitors were determined to be dependent upon the stabilization strength of the nonbonded interactions provided toward two catalytic residues: Arg55 and Asn102 in CypA and the analogous Arg63 and Asn110 residues in CypB. Fine-tuning of the hydrophobic interactions allowed for enhanced potency among derivatives. The aryl 1-indanylketones are predicted to differentiate between the cyclophilins by using distinct binding motifs that exploit subtle differences in the active site arrangements. Ideas for the development of new selective compounds with the potential for advancement to low-nanomolar inhibition are presented.