Slice Partially Diffusion Controlled Reactions

Unlike diffusion controlled reactions, where reaction takes place as soon as the interparticle separation hits the boundary a partially diffusion controlled reactions involve an extra complexity, such that the probability of reaction must be calculated based on the surface reactivity. This probability can be calculated by solving the backward diffusion equation to And the survival probability (a, B ) on going from boundary a to B (defined as a + 5) subject to a radiation boundary condition at surface a (situation 2 as shown in Fig. 4.7). Using the boundary condition (a) = v/D )p a) and Q. B ) = 1, the survival probability Q(a, B ) is found to be 

Although Eq. (4.122) allows the implementation of partially diffusion controlled reactions, another possibility needs to be considered on reaction at the inner boundary a the radical pair separates to x which is less than B (with B defined to be a+ 8) and survive reaction. The pair then undergo spin relaxation at the point x and re-approach the boundary a without ever hitting the boundary B. The likelihood of this situation can be tested using the recovering boundary formalism. A thorough derivation is now presented.  

The expression for the probability of going from atob without reaction is given by 

The first and second term in Eq. (4.123) can be written in terms of Laplace transform (with the Laplace variable a) as 

The transition density for a diffusion process subject to an inner reflective boundary and outer absorptive boundary can be found using the renewal theorem as 

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The last step to implementing partially diffusion controlled reactions into the Slice... 

There is obviously less diffusion control if the slices of immobilized enzyme are thin rather than thick, since the substrate then has ready access to the enzyme. Under biological conditions, substrate concentrations are usually substantially less than required to saturate the enzyme. Some diffusion control is therefore to be expected, especially if the macromolecular structural material is fairly thick. It has been estimated that in muscle filaments, of thickness approximately 0.1 micrometres (/ m), there is essentially no diffusion control. On the other hand with muscle fibers, of thickness approximately 5 xm, the enzyme reaction is almost completely diffusion controlled. Muscle fibrils, of thickness approximately 2 / m, lie in between, and there is partial diffusion control. 

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