Bulk substrate concentration

Figure 11.19 Plots of the external effectiveness factor as a function of the substrate modulus Da for different values of the dimensionless bulk substrate concentration is the limiting first-order effectiveness factor attained at sufficiently low concentrations. Adapted from C.Horvath and J.M.Engasser. Biotechnol.Bioeng., 16, 909 (1974).

Figure 11.20 Schematic plot of the overall rate of reaction catalyzed by a surface-bound biocatalyst against the bulk substrate concentration.

As can be seen in Figure 11.20, at high bulk substrate concentrations when the reaction is zero order, s will always approach and the reaction is kinetically controlled. At lower bulk substrate concentrations the reaction can be both kinetically or diffusionally controlled depending on the ratio of k. A and When k. A v, /K, mass... 

Thus the invocation of the steady-state assumption results in a rate law involving two independent dimensionless parameters C/KM, which compares the bulk substrate concentration with the Michaelis constant for the enzyme, and k3CEd2/DKM, which in effect compares the rate of catalysis with the rate of diffusion. These two parameters may be varied to show their effect on the detected current, and the reader is directed to the cited papers to view the results of this simulation. What is intended here, however, is a specific example of the use of the steady-state assumption in the development of an atypical rate expression (Eq. 20.102). 

Fig. 5.49. Dimensionless plot of overall reaction rate against bulk substrate concentration for a surface immobilised biocatalyst...

A plot of r t against the DamkOhler number is shown in Fig. 5.50 with the bulk concentration as parameter. On the graph it may be seen that r)e is dependent on both fib and Da, but that three identifiable regions exist. At low values of Da, kinetic control of the reaction is observed and the curves show that rje approaches unity for most substrate concentrations, whilst at high bulk substrate concentrations the... 

In the case of gel entrapped biocatalysts, or where the biocatalyst has been immobilised in the pores of the carrier, then the reaction is unlikely to occur solely at the surface. Similarly, the consumption of substrate by a microbial film or floe would be expected to occur at some depth into the microbial mass. The situation is more complex than in the case of surface immobilisation since, in this case, transport and reaction occur in parallel. By analogy with the case of heterogeneous catalysis, which is discussed in Chapter 3, the flux of substrate is related to the rate of reaction by the use of an effectiveness factor rj. The rate of reaction is itself expressed in terms of the surface substrate concentration which in many instances will be very close to the bulk substrate concentration. In general, the flux of substrate will be given by ... 

The effects of this concentration gradient are most significant at low bulk concentrations of the substrate, since substrate is converted to product as soon as it reaches the surface of the particle, so that the surface concentration of substrate is zero. At very high bulk substrate concentrations, the enzymatic reaction rate is limited by enzyme kinetics rather than mass transport, so that surface concentrations do not differ significantly from those in the bulk. Because of the concentration gradient, however, enzyme saturation with substrate occurs at much higher bulk substrate concentrations than required to saturate the soluble enzyme. Apparent Km values (K m) for immobilized enzymes are larger than Km values obtained for the native soluble enzymes. 

Figure 7.11 A theoretical potentiometric enzyme electrode calibration curve based on external diffusion control of the reaction a plot of the logarithm of the product concentration in the enzyme layer versus the logarithm of the bulk substrate concentration. K = 10" the value of kaEV/PsE is given on the curve [24],...

The surface concentration at the external surface of the pellet is equal to the bulk substrate concentration So... 

Fig. 4.7 Substrate conversion rate (r) as a function of bulk substrate concentration, showing limiting cases I and II (see text)...

A different approach is the direct determination of a and the intrinsic kinetic parameters from experimental rate data. This method was proposed by Chen (in Buchholz 1982) and is based on the determination of initial rates within a broad range of bulk substrate concentration. The kinetics of enzyme reaction, represented by the right-hand side of Eq. 4.14 is a very complex function of... 

Fig. 4.13 Global effectiveness factor (mean integral value) of a membrane immobilized enzyme with Michaelis-Menten kinetics as a function of bulk substrate concentration and Thiele modulus...

Formula (II.2.5) states that the NPV current is proportional to the bulk substrate concentration, C, the number of electrons transferred, n, the square root of the substrate diffusion coefficient, D, the electrode area. A, and is inversely proportional to the pulse time, tp. In fact, instead of the pulse time, the formula should contain... 

Figure 4.42. Graphical plot of the relationship between apparent overall rate coefficient /Capp and bulk substrate concentration Sl according to Equ. 4.104. Three distinct regions (A-C) correspond to the three cases in Fig. 4.41. (Reprinted with permission from Prog. Wat. Tech., vol. 12, Watanabe et al., copyright 1980, Pergamon Journals Ltd.)...

Since Xd can be calculated using Eqn. 11 and the current i and the bulk substrate concentration s° are generally known, then Eqn. 12 can be used to calculate the surface concentration s experimentally. 

FIGURE 2.3. Schematic representation of the substrate concentration profile (normalized with respect to bulk substrate concentration) as a function of distance from the surface of a rotating disk. The profile calculated via analytical solution of the convective diffusion equation, which is calculated from the Nernst diffusion layer approximation. Both calculations are shown. 

The situation becomes very simple if diffusional transport of the substrate in the solution can be neglected. The approximation is achieved by working with large bulk substrate concentrations, so there is very little depletion of substrate in the diffusion layer as a result of chemical reaction. In this case we can set s . = s , and Eqn. 34 becomes... 

The fact that catechol oxidation occurs via a Michaelis-Menten mechanism is confirmed by obtaining the RDE response for a range of catechol concentrations and subsequently subjecting the data to a Koutecky-Levich analysis [see (Fig. 2.36(a)]. The Koutecky-Levich intercept 7/cz, is then obtained and a Lineweaver-Burk analysis applied, which predicts that //fi, varies in an inverse manner with bulk substrate concentration according to... 

The concentration profile throughout the particle can be attained numerically by integrating Equation 4.57. The extent of mass transfer is commonly expressed by a local effectiveness factor, q, defined as the ratio between local reaction rates inside the catalyst particles and the reaction rate at the surface with bulk substrate concentration, as given by Eqnation 4.63. This can be rearranged to give Equation 4.64. The mean, or overall, effectiveness factor of the enzyme particles can be evaluated by an average integration of local effectiveness distribution inside the particle. 

Formula (II.2.5) states that the NPV current is proportional to the bulk substrate concentration, C, the number of electrons transferred, n, the square root of the substrate diffusion coefficient, D, the electrode area, A, and is inversely proportional to the pulse time, tp. In fact, the formula should contain the sampling time, ts y especially when the sampling is not performed exactly at the end of the pulse. Formula (II.2.5) is valid for electrodes of regular size (radius in the range of a few mm) and for processes where the transport of the substrate to the electrode surface is done only by diffusion. Chemical reactions involving the substrate that either precede or follow the electron transfer will also lead to different currents. The height of NPV waves does not depend on the electron transfer rate, so this technique is considered as a very reliable one for the determination of diffusion coefficients of the examined compounds. 

Let us take the simplifying approach that there is no external mass transfer limitation, as this approach will yield an upper limit on the chemical flux into a biofilm that is not limited by mass transfer outside the biofilm. To do that, we assume that the concentration at the biofilm surface is the bulk substrate concentration. 

You use an inunobilized enzyme system in which the enzyme is encapsulated in beads of radius of a polymer hydrogel at a mass loading density /Oe = 10 mg /ml of gel. The substrate diffiisivity in the hydrogel is 10 cm /s. The bulk substrate concentration is 1 M. Neglect external mass transfer resistance. Define an internal effectiveness factor, and compute its value as a function of in the range fO fO m. Since the enzyme is expensive, what radins wonld yon use, and why ... 

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