Fourier Least-squares Methods Research Articles

Approximate Periodic Solutions for Oscillatory Phenomena Modelled by Nonlinear Differential Equations

We apply the Fourier-least squares method (FLSM) which allows us to find approximate periodic solutions for a very general class of nonlinear differential equations modelling oscillatory phenomena. We illustrate the accuracy of the method by using several significant examples of nonlinear problems including the cubic Duffing oscillator, the Van der Pol oscillator, and the Jerk equations. The results are compared to those obtained by other methods.

Frequency-domain lifetime imaging methods at unilever research

Fluorescence lifetime imaging methodology has been successfully implemented at Unilever Research in a frequency-domain manner. The experimental rig constructed comprises a wide-bandwidth electrooptic modulator operating on a CW argon-ion laser. The modulated excitation with a typical upper modulation frequency limit of 200 MHz falls on macroscopic samples and the resultant scattered light or fluorescence emission is then imaged onto a custom gain-modulatable image intensifier and slow-scan CCD camera combination. Phase adjustment of the image intensifier relative to the laser modulator is achieved by the RF function generator driving the intensifier. Both homodyne and heterodyne (500-Hz) strobing modes are employed to generate a double image stack (scattered light reference and fluorescence emission) comprising an image sequence as a function of instrumental phase difference. These image stacks are analyzed by Fourier least-squares methods to yield lifetime images by both phase delay and normalized demodulation. Correct operation of the apparatus is deduced from the direct imaging of a quencher-induced lifetime variation of BODIPY disulfonate over a range of concentrations. A typical industrially relevant sample, comprising an investigation of the lifetime aspects of human dental enamel autofluorescence at 50MHz modulation frequency, is given. This shows that there are real emission lifetime decreases of about 0.5 nsec in white-spot lesion areas compared to the surrounding sound enamel.

Molecular-replacement studies on crystal forms of despentapeptide insulin

Despentapeptide insulin, prepared by removing the five C-terminal B-chain residues, occurs in several crystal forms. The relationship between these and the known pig 2-Zn insulin structure is analysed by the fast rotation function and R-factor searches. Reasonably good models were obtained even in the cases when there are two molecules in an asymmetric unit. The 2-Zn insulin coordinates were positioned in the C2 sheep cell and refinement commenced using the fast Fourier least-squares method. The packing of this modified insulin is quite different from that of all other insulins studied so far but the molecular structure is similar to that of 2-Zn insulin.

Manganese and calcium binding sites of concanavalin A

The atomic co-ordinates of concanavalin A, a lectin which specifically binds saccharides containing mannosyl and glucosyl residues, have been refined with the fast Fourier least-squares method of Agarwal (1978), with X-ray crystallographic data to 1.75 Å. This metalloprotein contains six co-ordinated Mn 2+ and seven co-ordinated Ca 2+ which are 4.25 Å apart and are close to the carbohydrate binding site. Both ions are pseudo-octahedral. This is possible for calcium since the average position of two oxygens (from the same carboxyl group) forms one apex of the octahedron. The refinement has improved this region considerably. The ligands for Mn 2+ are five oxygens, three from carboxyl side-chains and two from water molecules, and one nitrogen from a histidyl side-chain. The co-ordination sphere for Ca 2+ contains seven oxygens, three from two carboxyl side-chains, one from a peptide carbonyl, one from an amide side-chain and, as does Mn 2+, two from water molecules. The averages for the five Mn 2+O and the seven Ca 2+O bonds are 2.28 ± 0.04 (standard deviation) and 2.45 ± 0.09, respectively. For Mn 2+, this average is 0.10 Å higher than that found for the Mn(II)hexa-aquoion and is toward the high end of the range usually found for Mn 2+O complexes. This is of particular significance since Mn 2+ is used as a probe for nuclear magnetic resonance spectroscopic studies of native Mn 2+- and Mn 2+-substituted metalloproteins. Also, refinement confirms the non-proline cis peptide bond, which is immediately adjacent to the Ca 2+ and the carbohydrate binding sites. For the final model, the bond lengths and angles for backbone atoms have root-mean-square deviations from standard values of 0.030 Å and 5.2 deg., and side-chain atoms have slightly lower values. The crystallographic residual, R factor, for this model is 0.167 for 22,042 structure factors.

Refinement of concanavalin A with the fast Fourier least-squares method

Refinement of concanavalin A with the fast Fourier least-squares method

Crystallographic refinement of the structure of actinidin at 1.7 Å resolution by fast Fourier least-squares methods

The structure of the proteolytic enzyme, actinidin, has been refined by fast Fourier least-squares methods [Agarwal (1978), Acta Cryst. A34, 791-809]. Atomic positions were refined independently by the least-squares program, with the whole protein structure being regularized at intervals. After an initial refinement phase with an overall temperature factor, B, only, individual isotropic B values for all atoms were also refined. Overall, the crystallographic R factor was reduced from 0.429 (for 14 800 reflections to 2.0 Å resolution) to 0.171 (for all 23 990 reflections between 10 and 1.7 Å resolution), with a final estimated accuracy in atomic positions of <0.1 Å. The final model comprises 1657 protein atoms, constrained close to standard geometry, and 163 solvent molecules, the latter identified using somewhat selective criteria. Most of the structure refined automatically with an average shift of 0.45 Å for main-chain atoms and 0.56 Å for side-chain atoms (maximum shift about 1.5 Å). Some larger shifts resulted from manual intervention. Groups of atoms with high B values, or which were not refining well, were removed at intervals for scrutiny in difference maps, and major corrections were made to the conformations of 16 side chains and two peptide units. One correction to the amino-acid sequence was made (Asp 86 → Glx 86) and disordered conformations were introduced for five side chains. The whole refinement was completed in three months.

Structural relationships in the two-zinc insulin hexamer.

The refinement of the crystal structure of two-Zn pig insulin using 1.5-A (1A = 0.1 nm) resolution data by Fourier and fast Fourier least-squares methods allows us to make detailed comparisons between the two independent molecules present in the two-Zn insulin dimer and to describe their interactions in the monomer, dimer, and hexamer. The main chain structures for the two molecules agree well except at the N terminus of the A chain and the C terminus of the B chain. The residues along the line of the local two-fold axes, apart from the B25 side chain, conform extremely closely to the two-fold symmetry, although the discrepancies are much more apparent away from this axis. The ability of the insulin molecule to adopt different conformations may be an important factor in the expression of its biological activity.