Dimensionality, compartmentalized systems

Chawla MK, Lin G, Olson K, Vazdarjanova A, Burke SN, et al. 2004. 3D-catFISH a system for automated quantitative three-dimensional compartmental analysis of temporal gene transcription activity imaged by fluorescence in situ hybridization. J Neurosci Meth 139 13-24. 

Section III focuses on problems of system topology and documents, using results obtained from a series of model calculations, how the separate influences of system size, dimensionality, and reaction pathway(s) can be disentangled, and the principal effects on reaction efficiency quantified. With these factors clarified. Section IV demonstrates how these trends and correlations change when a multipolar potential is operative between reaction partners, both confined to a compartmentalized system. More general diffusion-reaction systems are described in Section V, where effects arising from nonrandom distributions of reaction centers and, secondly, the influence of multipolar potentials in influencing catalytic processes in crystalline and semiamorphous zeolites are explored. The conclusions drawn from these studies are then summarized in Section VI. 

The timing and efficiency of a diffusion-controlled process in a compartmentalized system can be enhanced by reducing the dimensionality... 

Having confirmed that the concept of reduction of dimensionality can play an important role in determining the efficiency of diffusion-controlled reactions in both symmetrical and asymmetrical compartmentalized systems, one may ask How does a substrate know, upon first encounter with the boundary, to move along the interior surface of a cellular unit, to react (eventually) with (say) a membrane-bound enzyme While substrate-specific, surface binding or association forces can conspire to keep the substrate in immediate vicinity of the boundary, once the latter has been encountered for the first time, certain (chemically) nonspecific, statistical factors are also likely to play a role in reduction of dimensionality. 

Figure 4.23. Cross-over in reaction efficiency as a function of system geometry for M X M X N lattices. The vertical axis calibrates the eccentricity s = N/M and the horizontal axis calibrates the surface-to-volume ratio S/V (see text). To the right of the hatched area, random d = 3 diffusion to an internal, centrosymmetric reaction center in the compartmentalized system is the more efficient process. To the left of the hatched area, reduction of dimensionality in the d = 3 flow of the diffusing coreactant to a restricted d = 2 flow upon first encounter with the boundary of the compartmentalized system is the more efficient process. The lines delimiting the hatched region give upper and lower bounds on the critical crossover geometries.

Regarding the dependence of the reaction efficiency on the dimensionality of the compartmentalized system, the studies reported in Sections III.B.3 and III.B.4 on processes taking place on sets of fractal dimension showed that, consistent with the results found for spaces of integer dimension, the higher the dimensionality of the lattice, the more efficient the trapping process, ceteris paribus. Processes within layered diffusion spaces, which can be characterized using an approach based on the stochastic master equation (4.3), show a gradual transition in reaction efficiency from the behavior expected in c( = 2 to that in = 3 as the number k of layers increases from fe = 1 to k = 11. 

Studies of tissue effects are usually based on animal (in vivo) experiments. Because of ethic reasons, and the considerable efforts to perform animal experiments, several experimental In vitro systems have been developed to mimic typical tissue effects as closely as possible. These comprise comparably simple systems based on a three-dimensional growth pattern distinct from the conventionally used monolayers of cells [56-58] up to quite complex systems with coculture of different cell types or utilizing the differentiation capacity of cells in vitro, leading to highly structured and compartmentalized 3D-cell systems [59]. 

A similar procedure can be applied to the device we call a one-dimensional (1-D) pipe, shown in Figure 2.1b. In this configuration, the properties of the system are time invariant, but vary with distance in the direction of flow. The system is said to be at steady state and distributed in one spatial coordinate. This is the exact reverse of the conditions that prevailed in a compartment. The physical phenomena that take place in the two cases, however, are similar. Mass is again transported by bulk flow, enters and leaves the device by exchange with the surroundings, and is generated or consumed by chemical reaction. The only difference here is that mass can also enter and leave by diffusion, which was not the case in the compartmental model. 

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