Front Matter

Diffusion limitation

An important quality criterion of a carrier applied to enzyme immobilization is its

protein capacity. Thus, porous materials are mainly used for this purpose and the

kinetics of biocatalysts immobilized to such carriers by adsorption or covalent bind-

ing or by entrapment within a polymer matrix is often influenced by diffusion effects.

Such effects are subdivisible into two types: theexternaland theinternaldiffusion

barrier. The theoretical treatment of this type of heterogeneous catalysis profited

greatly from the large amount of knowledge acquired in connection with research

on catalysts for chemical process engineering. Important parameters such as the

Damko ̈hler numberU^2 , the Thiele modulusu(u¼(U^2 )0.5), and effectiveness factor

g(Damko ̈hler, 1937; Thiele, 1939) characterize the mass transfer in technical as well

as in natural catalyst systems. The effectiveness factor is expressed byg¼ri/r 0 in

whichriandr 0 are the measured reaction rate in the presence of the immobilized

biocatalyst, and the reaction rate in solution for equal enzyme concentrations and

reaction conditions. The external diffusion limitation arises from the fact that a par-

ticle in solution is surrounded by the so-called unstirred Nernst layer with the thick-

nessdthrough which the substrate molecules have to penetrate by diffusion towards

the particle surface. As a consequence of the resulting concentration gradient, the

immobilized catalyst will not be faced with the same substrate concentration as

it would be in solution under the same conditions; hence the measured reaction rates

will be different. Asddepends on the relative movement of the catalyst-containing

particles to that of the reaction medium, the mass transfer resistance can be reduced

by enhancing the stirring rate in a continuous-stirred tank reactor (or the bulk liquid

flow rate in a packed-bed reactor).

If the diffusion coefficient for the substrate molecules in solution (D 0 ) is higher

than that within the porous support (Deff), a concentration gradient is formed so that

the substrate concentration decreases with increasing distance from the surface of the

particle, and the enzyme-catalyzed reaction slows down. In case of strong internal

diffusion limitation and a large particle diameter 2Rthe substrate may no longer

reach the enzyme molecules in the region of the particle center. Mathematical treat-

ment of the substrate diffusion process (using Fick’s second law) and the catalytic

reaction running in parallel (under the assumption that the Michaelis – Menten me-

chanism is valid, and that the system is at steady state) leads to an expression contain-

ing the Damko ̈hler numberU^2 ¼L^2
V’max/(Deff
K’m) whereV’maxandK’mare the

maximum reaction rate (or density of biocatalyst) and the Michaelis – Menten con-

stant within the matrix, andLis the thickness of a membrane. In case of a spherical

bead with the diameter 2R,Lis replaced byR/3. The Damko ̈hler number is repre-

senting the relation of a characteristic reaction rateV’max/K’mto a characteristic

diffusion rateDeff/L-2. The largerU^2 (oru) the steeper the substrate concentration

decreases within the catalyst bead. The equation also contains information of how to

diminish the mass transfer resistance and thus also of how to increaseg. Figure 5a

and b show to what extent the activity of an immobilized enzyme depends on the

particle size and on the protein loading (Grunwald, 2000). Apart from the particle

size and biocatalyst loading test an existing diffusion limitation can be recognized by

an enhancedKmvalue and a diminished activation energyEAdue to the fact that

diffusion-controlled processes merely reveal activation energies between 5 and

272 13 Preparation and Application of Immobilized Phospholipases