Watson sphere

How can Equation (11.79) be solved Before computers were available only simple ihapes could be considered. For example, proteins were modelled as spheres or ellipses Tanford-Kirkwood theory) DNA as a uniformly charged cylinder and membranes as planes (Gouy-Chapman theory). With computers, numerical approaches can be used to solve the Poisson-Boltzmann equation. A variety of numerical methods can be employed, including finite element and boundary element methods, but we will restrict our discussion to the finite difference method first introduced for proteins by Warwicker and Watson [Warwicker and Watson 1982]. Several groups have implemented this method here we concentrate on the work of Honig s group, whose DelPhi program has been widely used. 

E. 1. Newman and A. Watson, Microbial abundance in the rhizo.sphere a computer model. Plant Soil 48 1 (1977). 

In the region outside the sphere jn and yn are well behaved therefore, the expansion of the scattered field involves both of these functions. However, it is convenient if we now switch our allegiance to the spherical Hankel functions h[]) and h% We can show that only one of these functions is required by considering the asymptotic expansions of the Hankel functions of order v for large values of p (Watson, 1958, p. 198) ... 

An approach that is very closely related to the Atomic Sphere Approximation is the Renormalized Atom Theory, introduced first by Watson, Ehrenreich, and Hodges (1970) (sec also Watson and Ehrenreich, 1970, Hodges et al., 1972, and particularly Gelatt, Ehrenreich, and Watson, 1977). The name derives from the way the potential is constructed a charge density for each atom is constructed on the basis of atomic wave functions that are truncated at the Wigner-Seitz, or atomic, sphere. The charge density from each state is then scaled up (renormalized) to make up for that density beyond the sphere which has been dropped. 

Dotted lines represent pseudoknot Watson-Crick type interactions, b) Tertiary fold and c) three-dimensional topology in the crystal structure of the ribozyme - product complex. One enantiomer of the Diels-Alder product (sticks with transparent spheres) is bound into the catalytic pocket of the ribozyme ... 

The middle of the twentieth century marked the end of a long period of determining the building blocks of chemistry chemical elements, chemical bonds, and bond angles. The lists of these are not definitely closed, but future changes will be more cosmetic than fundamental. This made it possible to go one step further and begin to rationalize the strucmre of molecular systems, as well as to foresee the structural features of the compounds to be synthesized. The crucial concept is based on the Bom-Oppenheimer approximation and on the theory of chemical bonds and resulted in the spatial structure of molecules. The great power of such an approach was first proved by the construction of the DNA double helix model by Watson and Crick. The first DNA model was built from iron spheres, wires, and tubes. 

There has been very little theoretical and experimental work performed in this area owing to the complexity of such a system. Oman and Watson (1944) conducted experiments using particles of various shapes and found the voidage of the packing increased in the following order cylinders, spheres, granules, Raschig... 

G. S. Watson, Statistics on Spheres, Wiley-Interscience, New York, 1983. 

In a classic paper, Miron, Watson and Hay ° presented experimental evidence that large amounts of the ammonium salts remain in the explosive residue left from exploded ammonium salt-explosive mixtures used as commercial explosives in a 3.7 meter diameter steel sphere with the inside surface metallized with aluminum. Up to 50% of the ammonium nitrate was recovered from the explosion residues of granular, water-gel, and gelatinous explosives while no oil was recovered. Large percentages of sodium chloride, sodium nitrate, calcium carbonate and even nitrocellulose were recovered from some of the commercial explosives. Up to 96% of the aluminum was recovered in the residue of an aluminum containing water-gel explosive. 

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