Reaction-diffusion cells

The previous section introduced the first situation in this book where spatial distribution has been relevant. Even so, the only transport process was that imposed by the fluid flow no molecular transport mechanisms were invoked, except perhaps through the hoped-for perfect radial mixing. 

In this section we mov6 on to examine the behaviour of an open system in which the transport of reactants and products relies on molecular diffusion processes. The situation we envisage is that of a reaction zone in which various species diffuse and react. Outside this zone there exists an external reservoir in which the reactants have fixed concentrations. The reservoir provides a source of reactants which can diffuse across the boundary into the reaction zone, and either a source or a sink for the intermediates and products. The reaction-diffusion cell is sketched in Fig. 9.2. 

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The previous chapters have discussed the behaviour of non-linear chemical systems in the two most familiar experimental contexts the well-stirred closed vessel and the well-stirred continuous-flow reactor. Now we turn to a number of other situations. First we introduce the plug-flow reactor, which has strong analogies with the classic closed vessel and which will also lead on to our investigation of chemical wave propagation in chapter 11. Then we relax the stirring condition. This allows diffusive processes to become important and to interact with the chemistry. In this chapter, we examine one form of the reaction-diffusion cell, whose behaviour can be readily understood by comparison with the responses observed in the CSTR. 

Fig. 9.2. A reaction-diffusion cell. The reactants and products, here denoted A and B, have constant concentrations a and beK in the surrounding reservoir, but these vary across the reaction zone, — a0 < r < + a0.

A solution of the reaction-diffusion equation (9.14) subject to the boundary condition on the reactant A will have the form a = a(p,r), i.e. it will specify how the spatial dependence of the concentration (the concentration profile) will evolve in time. This differs in spirit from the solution of the same reaction behaviour in a CSTR only in the sense that we must consider position as well as time. In the analysis of the behaviour for a CSTR, the natural starting point was the identification of stationary states. For the reaction-diffusion cell, we can also examine the stationary-state behaviour by setting doi/dz equal to zero in (9.14). Thus we seek to find a concentration profile cuss = ass(p) which satisfies... 

Fig. 9.3. Stationary-state concentration profiles aS5(p) for a reaction-diffusion cell with a single cubic autocatalytic reaction (a) D = 0.1157, only small extents of reactant consumption arise (b) D = 0.0633, a higher extent of reactant consumption occurs, particularly towards the centre...

FIG. 9.6. The five qualitative forms for the stationary-state locus ass(0)-D for cubic autocatalysis with decay in a reaction-diffusion cell (a) unique (b) single hysteresis loop (c) mushroom (d) isola (e) isola + hysteresis loop, (f) The division of the Pcx-k2 parameter plane giving the five regions corresponding to the stationary-state forms in (a)-(e) note that the region for response (e), shown inset, is particularly small and has not yet been successfully located. 

So far almost all aspects of the stationary-state and even the time-dependent behaviour of this reaction-diffusion system differ only qualitatively from that found in the corresponding CSTR. In this section, however, we can consider a variation for which there can be no parallel in the well-stirred system—that of a reaction-diffusion cell set up with asymmetric boundary conditions. Thus we might consider our infinite slab with separate reservoirs on each side, with different concentrations of the autocatalyst in each reservoir. (For simplicity we will take the reactant concentration to be equal on each side.) Thus if we identify the reservoir concentration for p < — 1 as / L and on the other side (p + 1) as / R, the simple boundary conditions in eqn (9.11) are replaced by... 

The ratio of the overall rate of reaction to that which would be achieved in the absence of a mass transfer resistance is referred to as the effectiveness factor rj. SCOTT and Dullion(29) describe an apparatus incorporating a diffusion cell in which the effective diffusivity De of a gas in a porous medium may be measured. This approach allows for the combined effects of molecular and Knudsen diffusion, and takes into account the effect of the complex structure of the porous solid, and the influence of tortuosity which affects the path length to be traversed by the molecules. 

FIG. 6 Schematic of the rotating diffusion cell. The reaction is usually followed by sampling the bulk solution of the outer phase using a suitable analytical technique. 

Although reaction-diffusion limitation and the presence of nutritionally restricted phenotypes are obviously important determinants of biofilm drug resistance, neither, either separately or in combination, provides a complete explanation of the phenomena. Cells on the periphery of the biofilm, subject to nutrient fluxes similar to planktonic organisms would succumb to antibacterial concentrations that are effective against the planktonic cells. Cell-death at the periphery would lead to increased nutrient availability for deeper-lying cells. These would, in turn, grow faster and adopt a more susceptible... 

If nonreactive MPC collisions maintain an instantaneous Poissonian distribution of particles in the cells, it is easy to verify that reactive MPC dynamics yields the reaction-diffusion equation,... 

Schwann cells incorporated in a collagen matrix and injected into PLLA conduits were found to demonstrate comparable SFI values compared to isograft controls, but showed a statistically lower number of axons for both the high and low density Schwann cells groups and the collagen samples compared to the isograft controls (Evans et al., 2002). These results can be explained by a simple reaction diffusion model (Rutkowski and Heath, 2002b) described earlier. 

Glade, N., Demongeot, J., and Tabony, J. (2004). Micrombules self-organization by reaction-diffusion processes causes collective transport and organization of cellular particles. BMC Cell Biol, 5, 5-23. 

Theories that account for pattern formation in a morphogenetic field, as a result of reaction-diffusion processes, must assume the existence of at least two small diffusable molecules throughout the field. These hypotheses can be relaxed if one considers that the concentration of morphogenetic substances is altered in each cell via nonlinear interactions between cell surface receptors. 

Two of the most fundamental problems in developmental biology are the manner in which cells in different regions of an embryo come to adopt different developmental programs, and the relation between such different programs. The purpose of this chapter is to indicate how bifurcations in reaction-diffusion systems may offer new insights into these problems. The discussion will center on the fruit fly, Drosophila melanogaster. 

Utilization of whole cells and tissues in biosensor has increasingly been used. Enzyme stability, availability of different enzymes and reaction systems, and characteristics of cell surface are the advantages of using cells and tissues in biosensor designs. Multi-step enzyme reactions in cells also provide mechanisms to amplify the reactions that result in an increase in the detectability of the analytes. The presence of cofactors such as NAD, NADP, and metals in the cells allows the cofactor-dependent reactions to occur in the absence of reagents. (34, 50, 69). However, the diffusion of analytes through cell wall or membrane imposes constraint to this type of biosensors and results in a longer response time compared to the enzyme biosensors. 

Patterns Generated by the Outer Membrane. As in the single cell case, the outer membrane itself can generate electro-physiological patterns, even in the absence of reactions in the cell mass, R = 0. Such surface induced patterns were first discussed in the context of reaction-diffusion patterns in Ref. 

Strong deviations from a Turing type reaction-diffusion theory are expected to occur when the pattern length is of order of the cell size, i.e. near the upper limit of (97). 

To solve highly nonlinear differential equations for systems far from global equilibrium, the method of cellular automata may be used (Ross and Vlad, 1999). For example, for nonlinear chemical reactions, the reaction space is divided into discrete cells where the time is measured, and local and state variables are attached to these cells. By introducing a set of interaction rules consistent with the macroscopic law of diffusion and with the mass action law, semimicroscopic to macroscopic rate processes or reaction-diffusion systems can be described. 

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