Three-dimensional superposition

The sequence conservation is reflected in a highly conserved secondary and tertiary structure that is most clearly illustrated in the three-dimensional superposition of C atoms. Ignoring the C-terminal domains of PVC and HPII, the deviation of C atoms in a superposition of HPII with PVC, BLC, PMC, and MLC results in root mean square deviations of 1.1,1.5,1.6, and 1.5 A for 525, 477, 471, and 465 eqiuvalent centers, respectively (83). In other words, there is very little difference in the tertiary structure of the subunits over almost the complete length of the protein. The large and small subunits are shown in Fig. 8 for comparison. 

A particularly advantageous feature of computer graphical methods is the chance of three-dimensional superposition of two molecules for structural comparison. For example, Freidinger et al. identified a close conformational similarity between type-II P-tums and segments in which the residues i + 1 and i + 2 were replaced... 

Three-dimensional superposition is an abstract task involving the placement of one object in the space of another. Computational techniques that superimpose molecules. serve... 

MJ Sutcliffe, I Haneef, D Carney, TL Blundell. Knowledge based modelling of homologous proteins. Part I Three dimensional frameworks derived from the simultaneous superposition of multiple structures. Protein Eng 1 377-384, 1987. 

Superposition of Flows Potential flow solutions are also useful to illustrate the effect of cross-drafts on the efficiency of local exhaust hoods. In this way, an idealized uniform velocity field is superpositioned on the flow field of the exhaust opening. This is possible because Laplace s equation is a linear homogeneous differential equation. If a flow field is known to be the sum of two separate flow fields, one can combine the harmonic functions for each to describe the combined flow field. Therefore, if d)) and are each solutions to Laplace s equation, A2, where A and B are constants, is also a solution. For a two-dimensional or axisymmetric three-dimensional flow, the flow field can also be expressed in terms of the stream function. 

The CED-3 sub-family shows very similar three-dimensional structures. Figure 7.6 shows the superpositions of the Ca atoms. The structural backbones of the proteins within the family are closely superposed. 

The calculation method and equations presented in the previous sections are for Newtonian fluids such that the flow due to screw rotation and the downstream pressure gradient can be solved independently, that is, via the principle of superposition. Since most resins are highly non-Newtonian, the rotational flow and pressure-driven flow in principle cannot be separated using superposition. That is, the shear dependency of the viscosity couples the equations such that they cannot be solved independently. Potente [50] states that the flows and pressure gradients should only be calculated using three-dimensional (3-D) numerical methods because of the limitations of the Newtonian model. 

Viswanadhan, V.N., Ghose, A.K., Revankar, G.R., and Robins, R.K. Atomic physiochemical parameters for three dimensional structure directed quantitative structure-activity relationships. 4. Additional parameters for hydrophobic and dispersive interactions and their application for an automated superposition of certain naturally occurring nucleoside antibiotics, / Chem. Inf. Comput. Scl, 29(3) 163-172, 1989. 

Fond et al. [84] developed a numerical procedure to simulate a random distribution of voids in a definite volume. These simulations are limited with respect to a minimum distance between the pores equal to their radius. The detailed mathematical procedure to realize this simulation and to calculate the stress distribution by superposition of mechanical fields is described in [173] for rubber toughened systems and in [84] for macroporous epoxies. A typical result for the simulation of a three-dimensional void distribution is shown in Fig. 40, where a cube is subjected to uniaxial tension. The presence of voids induces stress concentrations which interact and it becomes possible to calculate the appearance of plasticity based on a von Mises stress criterion. 

Sippl, M.J. and Stegebuchner, H. Superposition of three dimensional objects a fast and numerically stable algorithm for the calculation of the matrix of optimal rotation. Comput. Chem. 1991, 35, 73-78. 

Handschuh, S., Wagener, M., and Gasteiger, J. (1998) Superposition of three-dimensional chemical structures allowing the conformational flexibility by a hybrid method../. Chem. Inf. Comput. Sci. 38, 220-232. 

EXAMPLE 2.4 Two leaky barrels of toxic material buried deep under a lake (unsteady, three-dimensional solution with two pulses and superposition)... 

Viswanadhan, V. N., et al.. Atomic Physicochemical Parameters for Three Dimensional Structure Directed Quantitative Structure-Activity Relationships IV. Additional Parameters for Hydrophobic and Dispersive Interaction and Their Application for an Automated Superposition of Certain Naturally Occurring Nucleoside Antibiotics. J. Chem. Inf. Comput. Sci., 1989 29, 163-172. 

The Patterson function has been the most useful and generally applicable approach to the solution of the phase problem, and over the years a number of ingenious methods of unraveling the Patterson function have been proposed. Many of these methods involve multiple superpositions of ports of the map. or "image-seeking with known vectors. Such processes are ideally suited lo machine compulation. Whereas the great increase in the power of x-ray methods of structure determination in the past few years has come simply front our ability lo compute a three-dimensional Patterson function, it is reasonable lo expect that, as machine methods of unraveling the Patterson function are developed, this power will increase many fold. 

The crystal structure of 4(5)-nitro-5(4)-methoxyimidazole contains a 1 1 mixture of two tautomers, 4-nitro-5-methoxy- and 5-nitro-4-methoxyimidazole [55], This is one of the very few cases of 4,5-disubstituted imidazoles for which there are two annular tautomers in the crystal. The molecular structure is the superposition of these tautomer forms. The structure is centrosymmetric and the N-H hydrogen atoms are disordered over two ring N atoms. Owing to the hydrogen-bond pattern, the values of their site occupation factors have to be exactly equal to 1/2. The molecules are connected into a three-dimensional network by means of N-H... N and C-H... 0 hydrogen bonds [55],... 

Hansen and Pearson [27] have recently employed a linear superposition of exponential repulsive interaction potentials between an inert atom and the atoms of a homonuclear diatomic molecule. They then performed a three-dimensional semiclassical calculation of the vibrational-transition probability including simultaneous rotational transitions. They conclude that the effect of coupled rotational transitions leading predominantly to AJ = 2 affect the vibrational-transition rate by 50% or more. 

Figure 3 Three-dimensional structures in the crystal of Co -cobalamin ((5), (a)) and of coenzyme B12 ((3), (c)) according to X-ray analysis and superposition of the structure of the Co -corrin part of coenzyme (3) and that of the Co -corrin (5) (b)...

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