Enzyme Distribution Equations

Reversible Michaehs-Menten mechanisms provide us with another useful general relationships, the enzyme distribution equations. Since many enzymatic reactions are fully reversible, immediately after mixing the enzyme with the substrate, or vice versa, after mixing the substrate with the enzyme, the steady-state concentration of all forms of enzyme is established. In the reversible mechanism with one central complex, the enzyme is divided among two forms  

we can cast the enzyme distribution equations in terms of all individual rate constants for both kinetic mechanisms. In the reversible mechanism with one central complex, the enzyme distribution equations are 

In the reversible mechanism with two central complexes, the enzyme distribution equations are 

In this case, the denominator value should be taken from the rate law appropriate to this mechanism (Eq. 3.36). 

The distribution equations describe the distribution of enzyme among various possible forms. These distribution equations when multiplied by Eo give the steady-state concentrations of various enzyme forms. The distribution equations are very complex with bisubstrate and trisubstrate reactions. They have some inherent interest by themselves, however, and are useful in deriving rate equations for reactions in the presence of dead-end inhibitors (Chapter 11), and equations for rates of isotopic exchange (Chapter 16). 

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The fraction of each enzyme species in the steady-state mixture, the enzyme distribution equation, is the sum of all its products obtained from all partial patterns, divided by the sum of all products ... 

After all 16 rate constants were successfully identified, one can easily sort them out in the form of enzyme distribution equations ... 

Enzyme distribution equations for the Ordered Bi Bi system (Eq. (4.31)) are written in terms of kinetic constants and substrate concentrations. They aU have the same denominators as the rate equation and thus, if they are multiplied by the same factors given in Eq. (4.38), used to convert the rate equation into kinetic constant form, their denominators will also be expressed in terms of kinetic constants. The terms in the numerators of these equations represent entire denominator terms, which are easily expressed in terms of kinetic constants, except when some denominator terms are split between the numerators of two or more equations. There are two split denominator terms in Eqs. (4.31). The rate constants multiplying A5 and PQ in the numerator of the fEAB]/[Eo] distribution are only parts of (coefAB) and (coefPQ), respectively the other part of (coefAB) comes from the [EQ]/tEo] distribution, and the other part of (coefPQ) comes from the [ A]/[Eo] distribution. 

Now, we can leave the rapid equilibrium treatment and proceed with the King-Altman treatment. The rate constants afcg and pit6 in reaction (4.51) now represent the effective or apparent rate constants. The enzyme distribution equations are now ... 

If the rates of proton-transfer steps are high, in specific cases, the protons can be introduced into the rate equations for bisubstrate and even trisubstrate reactions in a straightforward manner, without a need for a complete derivation of rate equations (Schulz, 1994). If the protons are treated as dead-end inhibitors of enzymatic reactions, the concentration terms for protons can be introduced directly into the rate equations via the enzyme distribution equations, alleviating considerably the derivation procedure. In order to illustrate this type of analysis, consider the usual Ordered Bi Bi mechanism (Section 9.2). 

Table a. Numerator terms of enzyme distribution equations for e mechanism in c ion (i4<34)> bach numerator term is multiplied by a corresponding Michaelis pH biD( k>D. 

In order to introduce the terms for a dead-end inhibitor into the velocity equation, each enzyme distribution equation is multiplied by an appropriate Michaelis pH function. In order to do so, the corresponding distribution equations for the Ordered Bi Bi mechanism, found in Chapter 9 (Eq. (9.13)), were divided by Va, followed by a partial elimination ofiTeq- The entire procedure is shown in Table 2. 

Thus, with the aid of enzyme distribution equations, the rate equation with the pH dependent kinetic parameters (Eq. (14.35)) is transformed into a rate equation with pH independent kinetic parameters. 

Write a distribution equation for each enzyme form—for example, E/E0 = Ne/D, where Ne and D are the numerator (for E) and denominator terms, respectively. 

Assuming a uniform enzyme distribution within the cell volume at system start-up, the functional dependence of enzyme concentration in region A on the reaction time is obtained upon integration of the mass balance equation ... 

However, in steady-state systems, the relative concentration of a particular enzyme form may be expressed by several denominator terms, and, often, a particular denominator term represents more than one enzyme form. Generally, in steady-state systems, the denominator terms multiplied by the factor F are those which appear in the numerator of the distribution equation for the enzyme form combining with the inhibitor. If a distribution equation in terms of kinetic constants cannot be written for the enzyme form combining with the inhibitor, than Ki cannot be calculated kinetically. 

The rate equation is obtained in the following manner. In the rate equation in the absence of products (Eq. (9.15)), those terms in the denominator representing free enzyme are multiplied by ii+B/KO, where Jf is the dissociation constant of the EB complex. The terms which represent the free enzyme are found in distribution equations for this system in the absence of products, these terms are JCiAlifB and KaB (found in Eq. (9.13)). Thus, the velocity equation becomes... 

From Eqs. (13.15), one can calculate the distribution of the total enzyme represented by each of the four enzjmie complexes plus the free enz5mie. From the distribution equations, the rate equation for a tetrametic enz5mie can be derived ... 

Distribution equations for bisubstrate reactions in the steady state are often very complex expressions (Chapter 9). However, in the chemical equilibrium, the distribution equations for all enzyme forms are usually less complex. Consider an Ordered Bi Bi mechanism in reaction (16.12) with a single central complex ... 

Section 16.2.1). Equation (16.7) contains rate constants, the concentration of labeled reactant, the concentration of unlabeled substrates, and the enzyme form the labeled reactant reacts with. The concentration of this enzyme form must be expressed in terms of rate constants and the concentration of reactants. In chemical equilibrium, this expression is relatively simple (Eq. (16.8)). Under the steady-state conditions, when the concentration of reactants is away from equilibrium, this enzyme form must be replaced from the steady-state distribution equation, which is usually more complex (Eq. (9.13)). Therefore, the resulting velocity equations for isotope exchange away from equilibrium are usually more complex and, consequently, their practical application becomes cumbersome. 

The last equation in Table 4 for the Bi Bi Uni Uni Ping Pong mechanism (in the absence of C and R) is relatively long and complex. The right-hand portion of the denominator in this equation represents the reciprocal concentration of free enzyme in chemical equilibrium multiplied by [EJ. If the products C and R are present, the distribution equation for [E]/[Eo] would be more complex (Eq. (12.73)) consequently, the resulting full rate equation for the A - Q isotope exchange becomes rather cumbersome. 

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 ... 

If EHi and S combine rapidly, and if the conversion of EHiS to EHi + P is the slow step, one can write a rate law similar to the Michaelis-Menten equation. In this case, however, the enzyme is distributed into additional forms (including four nonproductive forms E, ES, EH2 and EH2S) ... 

Dekker et al. [170] studied the extraction process of a-amylase in a TOMAC/isooctane reverse micellar system in terms of the distribution coefficients, mass transfer coefficient, inactivation rate constants, phase ratio, and residence time during the forward and backward extractions. They derived different equations for the concentration of active enzyme in all phases as a function of time. It was also shown that the inactivation took place predominantly in the first aqueous phase due to complex formation between enzyme and surfactant. In order to minimize the extent of enzyme inactivation, the steady state enzyme concentration should be kept as low as possible in the first aqueous phase. This can be achieved by a high mass transfer rate and a high distribution coefficient of the enzyme between reverse micellar and aqueous phases. The effect of mass transfer coefficient during forward extraction on the recovery of a-amylase was simulated for two values of the distribution coefficient. These model predictions were verified experimentally by changing the distribution coefficient (by adding... 

The statistics of processes such as radioactive decay and emission of light that produce a flux of particles or distributive polymerase enzymes that add residues at random to growing polymer chains obey the Poisson distribution (see Chapter 14). The number of particles measured per unit time or the number of residues added to a particular chain varies about the mean value x according to equation 6.41. 

The activation to the attack of pyrophosphate is measured by the pyrophosphate exchange technique. The enzyme, the amino acid, and ATP are incubated with [32P]-labeled pyrophosphate so that /f,y-labeled ATP is formed by the continuous recycling of the E-AA-AMP complex. The complex is formed as in equation 7.13, and the reaction is reversed by the attack of labeled pyrophosphate to generate labeled ATP. This process is repeated until the isotopic label is uniformly distributed among all the reagents. 

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