published by the press syndicate of the university of cambridgeThe Pitt
This is the 2005 second edition of a highly successful and well-respected textbook on the numerical techniques used to solve partial differential equations arising from mathematical models in science, engineering and other fields. The authors maintain an emphasis on finite difference methods for simple but representative examples of parabolic, hyperbolic and elliptic equations from the first edition. However this is augmented by new sections on finite volume methods, modified equation analysis, symplectic integration schemes, convection-diffusion problems, multigrid, and conjugate gradient methods; and several sections, including that on the energy method of analysis, have been extensively rewritten to reflect modern developments. Already an excellent choice for students and teachers in mathematics, engineering and computer science departments, the revised text includes more latest theoretical and industrial developments.
A series of helical structures for gramicidin A, with alternating L and D residues, are characterized as to number of residues per turn, atoms in hydrogenbonded rings, and dihedral angles. Because of alternating peptide C-O directions, these helices are capable of forming head-to-head hydrogen-bonded dimers with the capacity of functioning as transmembrane channels. The dimers are characterized as to channel length, pore size, and expected ion selectivity.In a test of the proposed head-to-head association for channel formation, the malonyl dimer [NN'-(dideform yl gramicidin A)-malonamideJ was synthesized. The chemical and conformational integrity of the product was verified by nuclear magnetic resonance; in lipid bilayer studies, the dimer was found to be a potent mediator of ion conductance with the predicted concentration dependence.Thus, the results on malonyl gramicidin A prove headto-head association in formation of the transmembrane channel, and the results are consistent with the specific geometrical configuration involved in head-to-head dimerization of lr(L,D) helices. At this stage, the action of gramicidin A on membranes with lipid-layer thicknesses of 30 A or less can best be understood in terms of the 7r(LD) helix with 6.3 residues per turn.On the basis of the Pauling-Corey-Donohue postulates for polypeptide structure (1-4), the conformational energy diagrams for backbone dihedral angles of polypeptides (5, 6), and the characteristics of gramicidin A-mediated ion conductance across lipid bilayers (7, 8,*), coupled with the significant fact that the amino acids in this peptide are in alternating I-D sequence (9, 10), a left-handed 2r(LD) helix has been proposed for the gramicidin A transmembrane channel (11), which has the unusual property of being capable of head-to-head dimerization. The head-to-head hydrogen-bonded association of two helices is possible because the peptide C-O bonds alternate in direction, with orientations primarily parallel and antiparallel to the helix axis. In this paper we generalize the possibilities of head-to-head association of left-handed gramicidin A helices of the TK(LD) type and demonstrate, by examining the lipid bilayer activity and NMR spectrum of a derivative obtained by chemically coupling the a-amino moieties of two deformyl gramicidin A molecules (i.e., by forming the malonamide dimer), that the transmembrane channel is formed by head-to-head association.THE SET OF 7(L.D) HELICES The sequence of gramicidin A, HCO-iVal-Gly-L-Ala-Trp-D-Leu-i-Trp-NHCH2CH20H, with its alternating L and D residues, makes possible a set of helices that differ from all previously described helices in that the peptide C-O bond vectors alternate with components parallel and antiparallel to the axis of the helix. If we define the direction of the helix axis vector from the amino to the carboxyl terminus, the C-O moieties of the L residues in lr(LD) helices of gramicidin A have components parallel to this vector, whereas for the D residues, they are antiparallel to it. In the a-...
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