1. Immobilized enzyme system
Bioreactors Engineering
Enzymes
Immobilized enzyme system
Diffusional Limitations in Immobilized Enzyme Systems
Diffusional resistances depend on;
1. Nature of the support material ( porous , nonporous).
2. Hydrodynamical conditions surrounding the support material.
3. Distribution of the enzyme inside or on the surface of the supporting material.
Depending on the value of the Damkohler number (Da). The rate of enzymatic
conversion may be limited by diffusion of the substrate or reaction.
If Da >> 1 the diffusion rate is limiting.
If Da << 1, the reaction rate is limiting.
If Da= 1, the diffusion and reaction resistances are comparable.

2. Bioreactors Engineering
Enzymes
Diffusion effects in surface-bound enzymes on nonporous support materials
Assume a situation where
Enzymes are bound and evenly distributed on the surface of a nonporous support
material,
2. All enzyme molecules are equally active,
3. Substrate diffuses through a thin liquid film surrounding the support surface to reach
the reactive surface,
4. The process of immobilization has not altered
the protein structure,
5. The kinetic parameters (Vm, Km) are unaltered.
At steady state, the reaction rate is equal to the mass-transfer rate:

3. Bioreactors Engineering
Enzymes
Eq. above can be solved graphically as shown in Fig. such a plot also makes it easy to visualize the effects of parameter changes such as stirring rate, changes in bulk substrate concentration, or enzyme loading.
Curve A solution of the right side of the Eq.
Line B, is the mass transfer equation (left side).
The intersection of the two lines is the reaction rate ν, that can be sustained in the system.

4. Bioreactors Engineering
Enzymes
When the system is strongly mass-transfer limited, [Ss]=0, since the reaction is rapid
compared to mass transfer, and
the system behaves as pseudo first order.
When the system is reaction limited (Da<<1), the reaction rate is often expressed as;
When the system is strongly reaction limited,
[Sb] ≈ [Ss]
Where, with appropriate assumptions,
Under these circumstances, the apparent Michaelis-Menten constant is a function of
stirring speed.
Usually, is estimated experimentally as the value of [Sb], giving one-half of
the maximal reaction rate.

5. Bioreactors Engineering
Enzymes

6. Bioreactors Engineering
Enzymes

7. Bioreactors Engineering
Enzymes
End of Lecture-1

8. Bioreactors Engineering
Enzymes
Diffusion effects in enzymes immobilized in a porous matrix
When enzymes are immobilized on internal pore surfaces of a porous matrix, substrate diffuse through the tortuous pathway among pores and reacts with enzyme immobilized on pore surface. Diffusion and reaction are simultaneous in this case, see Fig. below;
Assume a situation where
Enzyme is uniformly distributed in a spherical support particle.
The reaction kinetics are expressed by Michaelis-Menten kinetics.
There is no partitioning of the substrate between the exterior and interior of the support.
Stating that at steady state diffusion rate is equal to reaction rate;
Effective diffusivity of substrate within the porous matrix.

9. Bioreactors Engineering
Enzymes
Diffusion effects in enzymes immobilized in a porous matrix
Eq. above can be written in dimensionless form by defining the following dimensionless variables: or
where
Eq. above can be numerically solved to determine the substrate profile inside the matrix. The rate of substrate consumption is equal to the rate of substrate transfer through the external surface of the support particle at steady state into the sphere.

10. Bioreactors Engineering
Enzymes
Diffusion effects in enzymes immobilized in a porous matrix
Under diffusion limitations, the rate per unit volume is usually expressed in terms of the effectiveness factor (ɳ) as follows:
The effectiveness factor (ɳ) is defined as the ratio of the reaction rate with diffusion limitation (or diffusion rate) to the reaction rate with no diffusion limitation.
The value of the effectiveness factor (ɳ) is a measure of the extent of diffusion limitation.
For ɳ < 1, the conversion is diffusion limited,
For ɳ≈1 values, conversion is limited by the reaction rate and diffusion limitation are negligible.
The factor is a function of ϕ and β as shown in Fig, below;

11. Bioreactors Engineering
Enzymes
Diffusion effects in enzymes immobilized in a porous matrix
The factor (ɳ) is a function of ϕ and β as shown in Fig, below;
For a zero-order reaction rate (β→0), ɳ≈1 for a large range of Thiele modulus values such as 1< ϕ <100.
For the first-order reaction rate (β→infininty), ɳ= (ϕ, β ) and ɳ is approximated to the following equation for the high values of ϕ.
Fig. Theoretical relationship between the effectiveness factor (ɳ) and first-order Thiele modulus (ϕ), for a spherical porous immobilized particle for various values of β (β is the dimensionless Michaelis constant (β=Km/Ss) ).

12. Bioreactors Engineering
Enzymes
Diffusion effects in enzymes immobilized in a porous matrix
when internal diffusion limits the enzymatic reaction rate, the rate-constants Vm,app and Km,app values are not true intrinsic rate constant, but apparent values.
To obtain true intrinsic rate constants in immobilized enzymes, diffusion resistances should be eliminated by using small particle sizes, a high degree of turbulence around the particles, and high substrate concentrations.

13. Bioreactors Engineering
Enzymes

14. Bioreactors Engineering
Enzymes
when internal diffusion limits the enzymatic reaction rate, the rate-constants Vm,app and Km,app values are not true intrinsic rate constant, but apparent values.

15. Bioreactors Engineering
Enzymes
End of Lecture-2