Enzyme design equation

A reaction which follows power-law kinetics generally leads to a single, unique steady state, provided that there are no temperature effects upon the system. However, for certain reactions, such as gas-phase reactions involving competition for surface active sites on a catalyst, or for some enzyme reactions, the design equations may indicate several potential steady-state operating conditions. A reaction for which the rate law includes concentrations in both the numerator and denominator may lead to multiple steady states. The following example (Lynch, 1986) illustrates the multiple steady states... 

This chapter describes the different types of batch and continuous bioreactors. The basic reactor concepts are described as well as the respective basic bioreactors design equations. The comparison of enzyme reactors is performed taking into account the enzyme kinetics. The modelhng and design of real reactors is discussed based on the several factors which influence their performance the immobilized biocatalyst kinetics, the external and internal mass transfer effects, the axial dispersion effects, and the operational stabihty of the immobilized biocatalyst. 

Furthermore, it can be shown that, in the limiting cases of first-order kinetics [Equation (11.35) also holds for this case] and zero-order kinetics, the equal and optimal sizes are exactly the same. As shown, the optimal holding times can be calculated very simply by means of Equation (11.40) and the sum of these can thus be used as a good approximation for the total holding time of equal-sized CSTRs. This makes Equation (11.31) an even more valuable tool for design equations. The restrictions are imposed by the assumption that the biocatalytic activity is constant in the reactors. Especially in the case of soluble enzymes, for which ordinary Michaelis-Menten kinetics in particular apply, special measures have to be taken. Continuous supply of relatively stable enzyme to the first tank in the series is a possibility, though in general expensive. A more attractive alternative is the application of a series of membrane reactors. 

The design equations described in the last section are only valid for ideal reactions when the reactions are kinetically controlled. However, the modelling of enzyme reactions should take into account several factors which influence their performance. These factors are ... 

These design equations for a free-enzyme batch reaction may be formulated in a similar manner for kinetic schemes other than the case used above. If, for example, the back reaction is significant then the procedure would make use of equation A5.4.13 to define 9t0, and at infinite time an equilibrium mixture of substrate and product would be obtained. 

With the enzyme gel concentration known, Equation 16 can be used as a design equation in order to estimate the amount of enzyme, N, required to build up an enzymatic gel layer of thickness x. 

As outlined above, enzymes (E) represent the best-known chemical catalysts, because they are uniquely designed to carry out specific chemical reactions in a highly efficient manner (5). They initially act by binding a substrate (S) to form an enzyme-substrate complex [E S], which undergoes specific chemistry (catalysis) to give the enzyme-product complex [E P], followed by dissociation of product (P) and free enzyme (E). Equation 5.1 represents a simplified version of this scenario ... 

Table 20.8 Design equations for enzyme-catalysed reactions in a PFR (or a BR with i replaced by t) and a CSTR (from Messing, 1975) ... 

When external or internal mass transfer resistances are negligible, ijg= 1 or i]i= 1, respectively. If intrinsic kinetic parameters (determined while using free enzymes or cells, with no mass transfer limitations) are known, the total effectiveness factor can thus be used together with the reactor design equations as... 

In the case of a packed bed reactor, a typical configuration for the use of inunobilized enzymes, where mass transfer resistances are not significant and Michaelis-Menten-type kinetics is observed, the reactor design equation is simply obtained from... 

Cellulase and all chemicals used in this work were obtained from Sigma. Hydrolysis experiments were conducted by adding a fixed amount of 2 x 2 mm oflSce paper to flasks containing cellulase in 0.05 M acetate buffer (pH = 4.8). The flasks were placed in an incubator-shaker maintained at 50 °C and 100 rpm. A Box-Behnken design was used to assess the influence of four factors on the extent of sugar production. The four factors examined were (i) reaction time (h), (ii) enzyme to paper mass ratio (%), (iii) amount of surfactant added (Tween 80, g/L), and (iv) paper pretreatment condition (phosphoric add concentration, g/L), as shown in Table 1. Each factor is coded according to the equation... 

When binding of a substrate molecule at an enzyme active site promotes substrate binding at other sites, this is called positive homotropic behavior (one of the allosteric interactions). When this co-operative phenomenon is caused by a compound other than the substrate, the behavior is designated as a positive heterotropic response. Equation (6) explains some of the profile of rate constant vs. detergent concentration. Thus, Piszkiewicz claims that micelle-catalyzed reactions can be conceived as models of allosteric enzymes. A major factor which causes the different kinetic behavior [i.e. (4) vs. (5)] will be the hydrophobic nature of substrate. If a substrate molecule does not perturb the micellar structure extensively, the classical formulation of (4) is derived. On the other hand, the allosteric kinetics of (5) will be found if a hydrophobic substrate molecule can induce micellization. 

A procedure that utilizes quantitative structure-activity relationships to assist in the characterization and design of enzyme alternative substrates, inhibitors, and effectors. In this procedure, the inhibition (or activity) results of a series of structurally related compounds are analyzed to determine the coefficients in an equation of the form ... 

Preliminary rate measurements should allow one to make a plot of initial velocity Vq versus [metal ion], and this should provide information on the optimal metal ion concentration. (For many MgATP -dependent enzymes, the optimum is frequently 1-3 mM uncomplexed magnesium ion.) Then, by utilizing pubhshed values for formation constants (also known as stability constants) defining metal ion-nucleotide complexation, one can readily design experiments to keep free metal ion concentration at a fixed level. To compensate properly for metal ion complexation in ATP-dependent reactions, one must chose a buffer for which a stability constant is known. For example, in 25 mM Tris-HCl (pH 7.5), the stability constant for MgATP is approximately 20,000 M Thus, one can write the following equation ... 

AGIRE computer program for, 249, 79-81, 225-226 comparison to analysis based on rates, 249, 61-63 complex reactions, 249, 75-78 experimental design, 249, 84-85 inhibitor effects, 249, 71-75 potato acid phosphatase product inhibition, 249, 73-74 preliminary fitting, 249, 82-84 prephenate dehydratase product inhibition, 249, 72-73 product inhibition effects, 249, 72-73 prostate acid phosphatase phenyl phosphate hydrolysis, 249, 70 reactions with two substrates, 249, 75-77 reversible reactions, 249, 77-78 with simple Michaelian enzyme, 249, 63-71 [fitting equations, 249, 63] with slow-binding inhibitors, 249, 88 with unstable enzymes, for kinetic characterization, 249, 85-89. 

Production of HA during the biosynthesis of the antibiotic nebularine (6) was demonstrated in another Streptomyces species. The formation of HA was confirmed both by chemical reactions designed to detect it and by MS analysis. An unusual enzymic deamination of adenosine was suggested, which resulted in release of HA, rather than of NH3, as a key step in the production of nebularine (equation 6). 

The first two sections of Chapter 5 give a practical introduction to dynamic models and their numerical solution. In addition to some classical methods, an efficient procedure is presented for solving systems of stiff differential equations frequently encountered in chemistry and biology. Sensitivity analysis of dynamic models and their reduction based on quasy-steady-state approximation are discussed. The second central problem of this chapter is estimating parameters in ordinary differential equations. An efficient short-cut method designed specifically for PC s is presented and applied to parameter estimation, numerical deconvolution and input determination. Application examples concern enzyme kinetics and pharmacokinetic compartmental modelling. 

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