Enzymes mass balances

When unstirred batch membrane units are used as reacting vessels, a steady state mass balance on retained species, i.e., enzymes, leads to the evaluation of their concentration as a function of the distance from the membrane surface, x. Taking into account both convective and diffusive mass transfer mechanisms, the enzyme mass balance equation at steady state is as follows ... 

This results in an apparent decrease in Vmax and an apparent increase in Ks. The rate equation for the formation of product, the dissociation constants for enzyme-substrate (ES and ESI) and enzyme-inhibitor (El and ESI) complexes, and the enzyme mass balance are, respectively. 

An expression for the concentration of the El complex can be obtained from the enzyme mass balance and dissociation constants ... 

The velocity of the reaction, equilibrium dissociation, and ionization constants for the different enzyme species, and enzyme mass balance are... 

The rate equation, steady-state MichaeMs constants, and enzyme mass balance for this mechanism are, respectively. 

The mechanism involved the overall conversion of [5] to [P], The reverse reaction is insignificant because only the initial velocity in one of the forward direction is concerned. The mass balance equation expressing the distribution of the total enzyme is ... 

The concentration of monomer present at any concentration of inhibitor is given by SC, and the concentration of dimer is given, considering mass balance, by (1 - 8)C. When an enzyme is treated simultaneously with two inhibitors, / and J, that bind in a mutually exclusive fashion, the fractional activity is given by (Copeland, 2000)... 

We begin by stating the two mass-balance equations that are germane to enzyme inhibitor interactions ... 

We start with two mass balance equations that describe the relationships between total, free and bound forms of the enzyme and ligand (inhibitor), respectively ... 

In many experimental situations one cannot easily determine the free concentrations of enzyme and inhibitor. It would be much more convenient to cast Equation (A2.5) in terms of the total concentrations of these two reactants, as these quantitites are set by the experimenter and thus known with precision. We can replace the terms for free enzyme and free inhibitor in Equation (A2.5) using the mass balance equations, Equations (A2.1) and (A2.2) ... 

Table 18.2 lists 30 of the molecules used in this study that are known to be substrates for active transport or active efflux. The mechanistic ACAT model was modified to accommodate saturable uptake and saturable efflux using standard Michaelis-Menten equations. It was assumed that enzymes responsible for active uptake of drug molecules from the lumen and active efflux from the enterocytes to the lumen were homogeneously dispersed within each luminal compartment and each corresponding enterocyte compartment, respectively. Equation (5) is the overall mass balance for drug in the enterocyte compartment lining the intestinal wall. 

The activation function of the biochemical neuron is defined by the reaction mechanism and the pertinent rate equations. This function is actually a set of differential equations derived from mass balances for the components taking part in the enzymic reactions in each biochemical neuron (see Section 4.1.3). 

The results actually showed a deracemization of the racemic hydroxyester 10 as opposed to enantioselective hydrolysis with formation of optically pure (R)-hydroxyester 10 and only 20 % loss in mass balance. Small quantities of ethyl 3-oxobutanoate 9 (<5%) were also detected throughout the reaction, leading the authors to suggest a multiple oxidation-reduction system with one dehydrogenase enzyme (DH-2) catalysing the irreversible reduction to the (R)-hydroxy-ester (Scheme 5). 

The direction of a reaction can be assessed straightforwardly by comparing the equilibrium constant (Keq) and the ratio of the product solubility to the substrate solubility (Zsat) [39]. In the case of the zwitterionic product amoxicillin, the ratio of the equilibrium constant and the saturated mass action ratio for the formation of the antibiotic was evaluated [40]. It was found that, at every pH, Zsat (the ratio of solubilities, called Rs in that paper) was about one order of magnitude greater in value than the experimental equilibrium constant (Zsat > Keq), and hence product precipitation was not expected and also not observed experimentally in a reaction with suspended substrates. The pH profile of all the compounds involved in the reaction (the activated acyl substrate, the free acid by-product, the antibiotic nucleus, and the product) could be predicted with reasonable accuracy, based only on charge and mass balance equations in combination with enzyme kinetic parameters [40]. 

Using this equation, together with a mass balance relationship ([E] = [E]t - [ES]), we can solve for [ES]/[E]t, the fraction of enzyme combined as enzyme-substrate complex (Eq. 9-18). 

Using the steady state assumption for the mechanism shown in Eq. 9-14, and writing a mass balance equation that includes not only free enzyme and ES but also El we obtain an equation relating rate to substrate concentration. It is entirely analogous to Eq. 9-15 but Km is replaced by an apparent Michaelis constant, K m ... 

In the derivation according to Michaelis and Menten, association and dissociation between free enzyme E, free substrate S, and the enzyme-substrate complex ES are assumed to be at equilibrium, fCs = [ES]/([E] [S]). [The Briggs-Haldane derivation (1925), based on the assumption of a steady state, is more general see Chapter 5, Section 5.2.1.] With this assumption and a mass balance over all enzyme components ([E]total = [E]free + [ES]), the rate law in Eq. (2.3) can be derived. 

The amount of immobilized protein can be determined by the mass balance between initial and final solutions [56], A combined qualitative method merged from the classical in-situ detection of enzyme activity and western blot analysis can be applied to determine the enzyme spatial distribution through the membrane thickness and along the membrane module and its activity after the immobilization [57-59]. 

Vectors, such as x, are denoted by bold lower case font. Matrices, such as N, are denoted by bold upper case fonts. The vector x contains the concentration of all the variable species it represents the state vector of the network. Time is denoted by t. All the parameters are compounded in vector p it consists of kinetic parameters and the concentrations of constant molecular species which are considered buffered by processes in the environment. The matrix N is the stoichiometric matrix, which contains the stoichiometric coefficients of all the molecular species for the reactions that are produced and consumed. The rate vector v contains all the rate equations of the processes in the network. The kinetic model is considered to be in steady state if all mass balances equal zero. A process is in thermodynamic equilibrium if its rate equals zero. Therefore if all rates in the network equal zero then the entire network is in thermodynamic equilibrium. Then the state is no longer dependent on kinetic parameters but solely on equilibrium constants. Equilibrium constants are thermodynamic quantities determined by the standard Gibbs free energies of the reactants in the network and do not depend on the kinetic parameters of the catalysts, enzymes, in the network [49]. 

One source of nonlinear compartmental models is processes of enzyme-catalyzed reactions that occur in living cells. In such reactions, the reactant combines with an enzyme to form an enzyme-substrate complex, which can then break down to release the product of the reaction and free enzyme or can release the substrate unchanged as well as free enzyme. Traditional compartmental analysis cannot be applied to model enzymatic reactions, but the law of mass-balance allows us to obtain a set of differential equations describing mechanisms implied in such reactions. An important feature of such reactions is that the enzyme... 

Applying mass-balance and thermodynamic constraints typically leaves one without a precisely defined (unique) solution for reaction fluxes and reactant concentration, but instead with a mathematically constrained feasible space for these variables. Exploration of this feasible space is the purview of constraint-based analysis. It has so far been left unstated that any application in this area starts with the determination of the reactions in a system, from which the stoichiometric matrix arises. This first step, network reconstruction, integrates genomic and proteomic data to determine carefully the enzymes present in an organism, cell, or subcellular compartment. The network reconstruction process is described elsewhere [107]. 

Treatment of coacervates with enzymes did not induce an interaction between coacervate drops. By using - C-labelled protein and performing a mass balance, we found that essentially none of the kappa-casein in gum arabic-kappa-casein coacervates was released by rennet hydrolysis. Pepsin did not appear to hydrolyze the protein in coacervates either. Apparently the proteins are sequestered from enzymic activity as a result of coacervate formation. 

Magnetic moment, 153, 155, 160 Magnetic quantum number, 153 Magnetization, 160 Magnetogyric ratio, 153, 160 Main reaction, 237 Marcus equation, 227, 238, 314 Marcus plot, slope of, 227, 354 Marcus theory, applicability of, 358 reactivity-selectivity principle and, 375 Mass, reduced, 189, 294 Mass action law, 11, 60, 125, 428 Mass balance relationships, 19, 21, 34, 60, 64, 67, 89, 103, 140, 147 Maximum velocity, enzyme-catalyzed, 103 Mean, harmonic, 370 Mechanism classification of. 8 definition of, 3 study of, 6, 115 Medium effects, 385, 418, 420 physical theories of, 405 Meisenheimer eomplex, 129 Menschutkin reaction, 404, 407, 422 Mesomerism, 323 Method of residuals, 73 Michaelis constant, 103 Michaelis—Menten equation, 103 Microscopic reversibility, 125... 

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