Twists and turns: a tale of two shikimate-pathway enzymes

We are studying two enzymes from the shikimate pathway, dehydroquinate synthase (DHQS) and 5-enolpyruvylshikimate-3-phosphate synthase (EPSPS). Both enzymes have been the subject of numerous studies to elucidate their reaction mechanisms. Crystal structures of DHQS and EPSPS in the presence and absence of substrates, cofactors and/or inhibitors are now available. These structures reveal movements of domains, rearrangements of loops and changes in side-chain positions necessary for the formation of a catalytically competent active site. The potential for using complementary small-angle X-ray scattering (SAXS) studies to confirm the presence of these structural differences in solution has also been explored. Comparative analysis of crystal structures, in the presence and absence of ligands, has revealed structural features critical for substrate-binding and catalysis. We have also analysed these structures by generating GRID energy maps to detect favourable binding sites. The combination of X-ray crystallography, SAXS and computational techniques provides an enhanced analysis of structural features important for the function of these complex enzymes.

Square planar mononuclear Pd(II) complexes of substituted 2-(pyrazole-1-yl)phenylamines with a helical twist

X-ray crystal structure shows that 3,5-dimethyl-1-(2-nitrophenyl)-1H-pyrazole (DNP) belongs to the rare class of helically twisted synthetic organic molecules. Hydrogenation of DNP gives 2-(3,5-dimethylpyrazole-1-yl)phenylamine (L) which on methylation yields [2-(3,5-dimethylpyrazole-1-yl)phenyl]dimethylamine (L'). Two Pd(II) complexes, PdLCl2 (1) and PdL'Cl-2 (2), are synthesized and characterized by NMR. X-ray crystallography reveals that 1 and 2 are unprecedented square planar complexes which possess well discernible helical twists. (C) 2007 Elsevier B.V. All rights reserved.

Integrating control systems defined on the frame bundles of the space forms

This paper considers left-invariant control systems defined on the orthonormal frame bundles of simply connected manifolds of constant sectional curvature, namely the space forms Euclidean space E-3, the sphere S-3 and Hyperboloid H-3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1, 3). Orthonormal frame bundles of space forms coincide with their isometry groups and therefore the focus shifts to left-invariant control systems defined on Lie groups. In this paper a method for integrating these systems is given where the controls are time-independent. In the Euclidean case the elements of the Lie algebra se(3) are often referred to as twists. For constant twist motions, the corresponding curves g(t) is an element of SE(3) are known as screw motions, given in closed form by using the well known Rodrigues' formula. However, this formula is only applicable to the Euclidean case. This paper gives a method for computing the non-Euclidean screw motions in closed form. This involves decoupling the system into two lower dimensional systems using the double cover properties of Lie groups, then the lower dimensional systems are solved explicitly in closed form.

Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations

In this paper, we extend to the time-harmonic Maxwell equations the p-version analysis technique developed in [R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version, SIAM J. Numer. Anal., 49 (2011), 264-284] for Trefftz-discontinuous Galerkin approximations of the Helmholtz problem. While error estimates in a mesh-skeleton norm are derived parallel to the Helmholtz case, the derivation of estimates in a mesh-independent norm requires new twists in the duality argument. The particular case where the local Trefftz approximation spaces are built of vector-valued plane wave functions is considered, and convergence rates are derived.