Dimensionality, compartmentalized systems reduction

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.

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