Internal diffusional restrictions

1 Models for Enzyme Kinetics Under Internal Diffusional Restrictions for Different Particle Geometries Effectiveness Factor 

The analysis will be done in three steps. In the fist step, differential equations will be developed by combining enzyme kinetics and mass transfer to obtain the substrate (and product) profile within the biocatalyst particle in the second step, local effectiveness factor profiles will be obtained from the previous step in the third step a global effectiveness factor will be obtained by adequately averaging that distribution. This global effectiveness factor describes the behavior of the biocatalyst particle as a whole and will be obtained in terms of measured and calculated parameters, being a useful way of incorporating IDR into enzyme reactor design and performance evaluation, as considered in section 5.3. 

Considering an enzyme membrane of width L, a material balance for substrate over the section of analysis of width Ax is  

Letting Ax — 0 in Eq. 4.34 and considering Michaelis-Menten intrinsic kinetics and steady state  

Differential Eq. 4.37 has no analytical solution for Michaelis-Menten intrinsic kinetics and has to be solved numerically considering the following boundary 

---

Combined Effect of External and Internal Diffusional Restrictions... 

Ishikawa H, Tanaka T, Kuro K et al. (1987) Evaluation of tme kinetic parameters for reversible immobihzed enzyme reactions. Biotechnol Bioeng 29 924-933 Jeison D, Ruiz G, Acevedo F et al. (2003) Simulation of the effect of intrinsic reaction kinetics and particle size on the behavior of immobihzed enzymes under internal diffusional restrictions and steady state operation. Proc Biochem 39(3) 393-399 Katchalski-Katzir E, Kraemer DM (2000) Eupergit C, a carrier for immobDization of enzymes of industrial potential. J Mol Catal B Enzym 10 157-176 Kheirolomoom A, Khorasheh F, Fazehnia H (2002) Influence of external mass transfer limitation on apparent kinetic parameters of peniciUin G acylase immobihzed on nonporous ultrafine silica particles. J Biosci Bioeng 93 125-129... 

Valencia, P., S. Flores, L. Wilson, and A. Illanes. 2011. Effect of Internal Diffusional Restrictions on the Hydrolysis of Penicillin G Reactor Performance and Specific Productivity of 6-Apa with Immobilized Penicillin Acylase. Applied Biochemistry... 

A reduced reaction rate may result from external diffusional restrictions on the surface of carrier materials. In stirred tanks external diffusion plays a minor role as long as the reaction liquid is stirred sufficiently. Further, partition effects can lead to different solubilities inside and outside the carriers. Partition has to be taken into account when ionic or adsorptive forces of low concentrated solutes interact with carrier materials [81 - 83]. The most crucial effects are observed in porous particles due to internal or porous diffusion as outlined below. 

Fig. 4.6 Schematic representation of external (EDR) and internal (IDR) diffusional restrictions...

Mass transfer limitations can be relevant in heterogeneous biocatalysis. If the enzyme is immobilized in the surface or inside a solid matrix, external (EDR) or internal (IDR) diffusional restrictions may be significant and have to be considered for proper bioreactor design. As shown in Fig. 3.1, this effect can be conveniently incorporated into the model that describes enzyme reactor operation in terms of the effectiveness factor, defined as the ratio between the effective (or observed) and inherent (in the absence of diffusional restrictions) reaction rates. Expressions for the effectiveness factor (rj), in the case of EDR, and the global effectiveness factor (t ) for different particle geometries, in the case of IDR, were developed in sections 4.4.1 and 4.4.2 (see Eqs. 4.39-4.42,4.53,4.54,4.71 and 4.72). Such functions can be generically written as ... 

Photolyses of 31-34 in homogeneous solution results in the formation of diphylethanes 39 (5-15%), phenols 38 (5-15%), ortho-hydroxyphenone 36 (40-60%), and para-hydroxyphenones 37 (20-25%). Small amounts of phenyl benzyl ether 35 (3-8%) were also detected. However, photolyses of all of the four esters on NaY zeolite and Nafion only produce ortho rearrangement products 36. Molecular models suggest that esters 31-34 can enter into NaY zeolite internal surface and the inverse micelle of Nafion. We believe that the preference for formation of ort/zo-hydroxyphenones 36 in the products is a consequence of the restriction on diffusional and rotational motion of the geminate radical pair. 

The description of the internal motion of the epoxypropyl ring of 23 is strongly model-dependent. This motion can be satisfactorily approximated either by free rotation about the C-5—C-6 bond or by a jumping process between two stable conformations. Discrimination between these two models from the relaxation data was not possible owing to a fortuitous similarity in the activation energies ( 17 kJ/mol) of the internal and overall diffusional motions.13 Inspection of molecular models indicates, however, that the rotation of the epoxypropyl ring is not sufficiently constrained to justify restricted rotation about the C-5—C-6 bond. 

---