Spatial propagation

The essential step in (4.20) is the repeated application of the Hamiltonian. While the application of the potential is simply a multiplication, application of the kinetic energy operator, which for the linear triatomic molecule has the form 

The Fourier method is best suited to cartesian coordinates because the expansion functions QtkR/LR etlr Lr are just the eigenfunctions of the kinetic energy operator. For problems including the rotational degree of freedom other propagation methods have been developed (Mowrey, Sun, and Kouri 1989 Le Quere and Leforestier 1990 Dateo, Engel, Almeida, and Metiu 1991 Dateo and Metiu 1991). 

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The thermal model is that of Semenov. Space-averaged variables were used in order to minimize the mathematical difficulties and hence the model does not give an account of the spatial propagation of cool flames. The energy conservation equation used is... 

Apart from acoustic phonons, which account for heat transport in insulating media, propagation of vibrational energy is usually not considered in crystals, as the dispersion of optical modes is normally very small over the Brillouin zone. However, there is an important class of optical vibrations in crystals for which spatial propagation can be the dominant property at optically accessible wave vectors. This class is identical with that of infrared active modes and its members are known as phonon-polaritons. ... 

The spatial propagation of Ca waves has long been observed in a variety of egg types after fertilization (Gilkey et al, 1978 Jaffe, 1983,1991, 1993 Busa Nuccitelli, 1985). In these cells, waves of Ca propagate over the cortex, from the site of fertilization. More recently, the wavelike propagation of Ca signals has been observed in other cells in... 

The propagation of concentric and spiral waves of Ca in Xenopus oocytes has also been studied (Lechleiter et al., 1991) by means of simulations based on cellular automata (Gerhard, Schuster Tyson, 1990 Markus Hess, 1990) such a mathematical representation of excitable systems in terms of a set of rules simulated on a computer considers the existence of a finite number of cell states (excitable, excited, refractory). The comparison of numerical simulations with experiments suggested (Lechleiter et al, 1991) that the species responsible for the spatial propagation of the wave is cytosolic rather than IP3. Moreover, the characteristics of the phenomenon fit with the view (Berridge Irvine, 1989) that CICR is the primary mechanism imderlying Ca " wave propagation. 

Fig. 9.30. Spatial propagation of a sharp Cef front of the type seen in eardiae cells (type 1 wave). Shown are six successive stages of the transient pattern obtained by numerical integration of eqns,(9.11) of the model based on CICR, from which the term Vj/S related to stimulation has been removed and to which the diffusion of cytosolic Ca has been added. In these simulations, the Ca -sensitive Ca pool is assumed to be distributed homogeneously within the cell. The latter is represented as a two-dimensional mesh of 20 x 60 points and diffusion is approximated by finite differences boundary conditions are of the zero-flux type. The terms related to influx from (vq) and into kZ) the extracellular medium only appear in the points located on the borders of the mesh. The diffusion coefficient of is equal to 400 pmVs other parameter...

Fig. 9.31. Spatial propagation of a Ce tide resembling the waves seen in hepa-tocytes, oocytes, or endothelial cells (type 2 wave). The transient pattern is obtained as in fig. 9.30 for parameter values yielding oscillations of a period of the order of 1 min Vq =-1.68 jiM/min, = 93 xM/min, - 500 p,M/min, Kf = 0.66 pM, 2 = 11 -M, k -16.8 min, A , = 1 min other parameter values are as in fig. 9.10. The spatial mesh contains 30 x 30 points (similar results are obtained with a mesh of 60 x 60 points). The black bar in the upper, left part denotes the initial, transient stimulation, which consists in raising locally the level of cytosoUc Ca to 1.5 pM at the left extremity while the rest of the cell is in the resting level of 0.1 pM the scale of Ca concentration extends from 0 (white) to 1.5 pM (black) (Dupont Goldbeter, 1992b, 1994).

In homogeneous reactions no new structures are formed except, of course, the molecules of the product themselves. If, however, new phases are produced, especially new solid phases, their establishment demands a spatial propagation of the reaction. 

As the initiator molar absoprtivity increases, so does the maximum initiation rate, the breadth of the propagating front decreases, and the rate of spatial propagation through the sample decreases. 

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