Conformability Maps

Packer M J, M P Dauncey and C A Hunter 2000. Sequence-dependent DNA Structure Dinucleotide Conformational Maps. Journal of Molecular Biology 295 71-83. 

Both equation 11 and the two-dimensional counterpart of equation 9 can be solved by several standard mathematical techniques, one of the more useful being that of conformal mapping. A numerical solution is often more practical for compHcated configurations. 

Conformal Mapping Every function of a complex variable w = f z) = u x, y) + iv(x, y) transforms the x, y plane into the u, v plane in some manner. A conformal transformation is one in which angles between curves are preserved in magnitude xnd sense. Every analytic function, except at those points where/ ( ) = 0, is a conformal transformation. See Fig. 3-48. 

CM Becker. Geometrical versus topological clustering An insight into conformation mapping. Proteins 27 213-226, 1997. 

From the quality-eost arguments made in Seetion 1.2, it is possible to plot points on the graph of Oeeurrenee versus Severity and eonstruet lines of equal failure eost (% isoeosts). Figure 2.22 shows this graph, ealled a Conformability Map. Beeause of uneertainty in the estimates, only a broad band has been defined. 

The use of the Conformability Map in determining the potential eosts of non-eonformanee or failure should be assoeiated with the applieation of evaluating and eomparing different design sehemes in praetiee. The estimated failure eosts aet as a measure of performanee by whieh to make a justified seleetion of a partieular... 

The Conformability Map enables appropriate Cp values to be seleeted and through the link with the eomponent variability risks, and q, it is possible to determine if a produet design has eharaeteristies that are unaeeeptable, and if so what the eost eonsequenees are likely to be. The two modes of applieation are highlighted on Figure 2.23. Mode A shows that the quality loss assoeiated with a eharaeteristie at Cpk = 1 and FMEA Severity (S) = 6 eould potentially be 8% of the total produet... 

For each q and q risk value and the Severity Rating (S), a level of design acceptability is determined from where these values intersect on the Conformability Map. The symbols, relating to the levels of design acceptability, are then placed in the nodes of the Conformability Matrix for each variability risk which the failure mode is directly dependent on for the failure to occur. Once the level of design acceptability has been determined, it can then be written on the Conformability Matrix in the Comments section. Cpi values predicted or comments for suppliers can be added too, although predicted Cp values can also be written in the variability risks results table. 

The characteristic was analysed using CA and q was found to be 9. The values of (/m = 9 and S = 8 are found to intersect on the Conformability Map above the 10% isocost line. (If they had intersected between two isocost lines, the final isocost value is found by interpolation.) If there is more than one critical characteristic on the component, then the isocosts are added to give a total isocost to be used in equation 2.15. The total failure cost is determined from ... 

Following the eompletion of the variability risks table, a Conformability Matrix was produeed. This was used to relate the failure modes and their severity eoming out of the design FMEA to the results of the Component Manufaeturing Variability Risk Analysis. The portion of the matrix eoneerned with the moulded hub ean be found in Figure 2.34(d) and was eompleted using the Conformability Map. 

The CA methodology is useful in this respect. It is comprised of three sections the Component Manufacturing Variability Risks Analysis, the Component Assembly Variability Risks Analysis and the determination of the Effects of Non-conformance through the Conformability Map. 

Sinee the poorest performanee of the assembly distribution would oeeur when shifted, Cpk values rather than Cp values are better design targets. The Cp value ean be used as a target for the assembly based on the severity of applieation and minimum failure eost of 0.01% of the total produet eost as determined by the Conformability Map. If only eapable solutions are to be generated, whieh have a minimum proeess eapability index of Cp =1.33 (or 30 ppm) for both eomponent and assembly distributions, then the number of eomponents in the assembly staek ean be as low as three using the proposed statistieal model (Harry and Stewart, 1988). The overall requirement is... 

Stability in the frequency domain 6.4.1 Conformal mapping and Cauchy s theorem... 

The construction of the nanotube from a conformal mapping of the graphite sheet shows that each nanotube can have up to three inequivalent (by point... 

Fig. 2. Depiction of conformal mapping of graphene lattice to [4,3] nanotube. B denotes [4,3] lattice vector that transforms to circumference of nanotube, and H transforms into the helical operator yielding the minimum unit cell size under helical symmetry. The numerals indicate the ordering of the helical steps necessary to obtain one-dimensional translation periodicity.

The conformal mapping will transform this lattice operation B,v to a rotation of 1tt/N radians around the nanotube axis, thus generating a C,v subgroup. 

For the nanotubes, then, the appropriate symmetries for an allowed band crossing are only present for the serpentine ([ , ]) and the sawtooth ([ ,0]) conformations, which will both have C point group symmetries that will allow band crossings, and with rotation groups generated by the operations equivalent by conformal mapping to the lattice translations Rj -t- R2 and Ri, respectively. However, examination of the graphene model shows that only the serpentine nanotubes will have states of the correct symmetry (i.e., different parities under the reflection operation) at the K point where the bands can cross. Consider the K point at (K — K2)/3. The serpentine case always sat-... 

As an example of a nanotube representative of the diameters experimentally found in abundance, we have calculated the electronic structure of the [9,2] nanotube, which has a diameter of 0.8 nm. Figure 8 depicts the valance band structure for the [9,2] nanotube. This band structure was calculated using an unoptimized nanotube structure generated from a conformal mapping of the graphite sheet with a 0.144 nm bond distance. We used 72 evenly-spaced points in the one-... 

KC706 Inhibitor enzyme conformation MAP kinase p38a Inflammatory diseases Phase II... 

Gordon, L.M., Lee, K.Y.C., Lipp, M.M., Zasadzinski, J.A., Walther, F.J., Sherman, M. A., and Waring, A.J. Conformational mapping of the N-terminal segment of surfactant protein B in lipid using C-13-enhanced Fourier transform infrared spectroscopy. J. Peptide Res. 

Fig. 15. Conformational map of cyclohexane. The diagram represents a partial qualitative pictorial polar projection of the conformational globe of Pickett and Strauss (106) it may be completed by rotating around 120 and 240°, respectively. Relative potential energies are given (kcal mole-1 force field of ref. 19 reference chair conformation). The lines inside the six-membered rings represent mirror planes (solid) and twofold axes (dotted), respectively.

Fig. 16. Three dimensional conformational map of cyclohexane. The representation is analogous to that of Fig. 15 the third (vertical) coordinate is the potential energy. The given calculated potential energy differences (kcal mole-1) of the minima and transition states are drawn to scale. The interconnecting curves are drawn qualitatively they are merely meant to indicate the absence of intermediate further minima and maxima. See ref. 106 for details of analytical representations of conformational maps of cyclohexane...

Figure 4.2 Conformation map of cellobiose. Enclosed area defines allowed conformations in which there are no major conformational restrictions arising from interactions between non-bonded atoms.

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