Michaelis-Menten equation experiments

Note the close analogy with the Lineweaver-Burk form of the simple Michaelis-Menten equation. In a diagram representing MV against MX one obtains a line which has the same intercept as in the simple case. The slope, however, is larger by a factor (1 + YIK-) as shown in Fig. 39.17b. Usually, one first determines and in the absence of a competitive inhibitor (F = 0), as described above. Subsequently, one obtains A" from a new set of experiments in which the initial rate V is determined for various levels of X in the presence of a fixed amount of inhibitor Y. The slope of the new line can be obtained by means of robust regression. 

The kinetic data from phosphate [28] and proton [58] release experiments were analyzed quantitatively by means of a modified Michaelis-Menten equation assuming activation when only one substrate molecule, S, is bound and inhibition when two substrate molecules are bound to P-gp as described by the following scheme ... 

Fig. 28.4. Degradation of phenol by a consortium of methanogens, as observed in a laboratory experiment by Bekins et al. (1998 symbols), and modeled using the Michaelis-Menten equation (solid line). Inset shows detail of transition from linear or zero-order trend at concentrations greater than KAy to asymptotic, first-order kinetics below this level. Broken line is result of assuming a first-order rather than Michaelis-Menten law.

It is apparent that O Eq. 5 is a variation of the MichaeUs-Menten equation. The inhibitor data shown in O Figure 4-7 can instead be fitted to O Eq. 5, holding Km (and Vjnax) constant to their control values (5 pM and 20 nmol/min/mg, respectively). The curve obtained is identical to that fitted with the Michaelis-Menten equation (O Figure 4-7), but nonlinear regression now yields the information that K, of the inhibitor equals 40% of the concentration at which it was included in the assay to obtain the best-fit curve. In other words, if the concentration of inhibitor present in the experiment shown in O Figure 4-7 was 25 pM, the Ki for the inhibitor is 10 pM. 

Equation E3.5 in this experiment can be used to determine / values, but hyperbolic plots are obtained. Can you convert Equation E3.5 into an equation that will yield a linear plot without going through all the changes necessary for the Scatchard equation Hint Study the conversion of the Michaelis-Menten equation to the Lineweaver-Burk equation. 

The Michaelis-Menten equation represents a mechanistic model because it is based upon an assumed chemical reaction mechanism of how the system behaves. If the system does indeed behave in the assumed manner, then the mechanistic model is adequate for describing the system. If, however, the system does not behave in the assumed manner, then the mechanistic model is inadequate. The only way to determine the adequacy of a model is to carry out experiments to see if the system does behave as the model predicts it will. (The design of such experiments will be discussed in later chapters.) In the present example, if substrate inhibition occurs, the Michaelis-Menten model would probably be found to be inadequate a different mechanistic model would better describe the behavior of the system. 

This equation should look familiar, because it is functionally identical to the Michaelis-Menten equation of enzyme kinetics. This equation also should make clear the experimental design to be used in determining KD and Bmax using saturation isotherms. We have as the independent variable [E] and as dependent variable B. A successful experiment should permit the estimation of the two biologically meaningful constants KD and 5max. 

Kinetic experiments may be used for revealing the type of inhibition in enzymes. By inserting experimental data to the inverted Michaelis-Menten equation this gives straight-line plots (Lineweaver-Burk), which can be extrapolated to yield the characterizing constants of the enzyme. However, the Michaelis-Menten model cannot account properly for the kinetic properties of allosteric enzymes [34]. 

Although it is quite simple to set up an experiment to determine the variation of v with [5], the exact value of V,n,y is not easily determined from hyperbolic curves. Furthermore, many enzymes deviate from ideal behavior at high substrate concentrations and indeed may be inhibited by excess substrate, so the calculated value of cannot be achieved in practice. In the past it was common practice to transform die Michaelis-Menten equation (9) into one of several reciprocal forms (equations [10] and [11]), and either 1/v was plotted against 1/[S], or [S]/v was plotted against [S]. 

Generally, the relationship between the initial metabolic velocity and unbound drug concentration is described by the Michaelis—Menten equation [Eq. (6.27)] in in vitro experiments ... 

Fig. 3. Use of [7 P]GTP to follow GAP reaction. (A) Decrease in protein associated P induced by [325-724] AS API. Two progress curves at different [L8K]Arfl GTP concentrations are shown from an experiment in which [L8K]Arfl GTP was loaded with [7 P]GTP as described in the text. The labeled [L8K]Arfl, at final concentrations ranging from 0.5 ijlM to 10 iiM, was added to a reaction mixture containing 360 PA, 90 iiM PIP2, 0.1% (w/v) Triton X-100, and 1 nM [325-724] AS API. Samples of the reaction were taken at the indicated times following addition of [L8K]Arfl GTP and protein bound radiolabel was determined. Binding stoichiometries were determined by measuring the amount of bound nucleotide of a known specific activity. The binding stoichiometry was used to calculate the amount of Arfl GTP present. (B) Replot of initial rate data. The initial rates of loss of protein-associated P, taken to be vi, were determined from the plots in (A). The determined vis were plotted against Arfl GTP concentration. The data were fit to the Michaelis-Menten equation.

Fig. 3. PfdUTPase-catalyzed hydrolysis of dUTP in 25 mM MES, 100 mM NaCl, 25 mM MgCl2,1 mM P-mercaptoethanol at pH 7 and 25 °C. (A) Typrical calorimetric trace (peal/ s versus time) obtained after addition of three injections of 5 mM dUTP (20 pi) to the calorimetric cell containing PfdUTPase (5.3 nM). (B) dUTP dilution thermogram. (C) Calorimetric trace (first injection) resulting after subtracting the first peak of the dilution experiment. (D) Net thermal power was converted to rate and fitted to the Michaelis-Menten equation, giving AHobs = 10.4 kcal/mol, = 3.2 pM, kcai = 11.7 s-i.

The efficiency of solution-phase (two aqueous phase) enzymatic reaction in microreactor was demonstrated by laccase-catalyzed l-DOPA oxidation in an oxygen-saturated water solution, and analyzed in a Y-shaped microreactor at different residence times (Figure 10.24) [142]. Up to 87% conversions of l-DOPA were achieved at residence times below 2 min. A two-dimensional mathematical model composed of convection, diffusion, and enzyme reaction terms was developed. Enzyme kinetics was described with the double substrate Michaelis-Menten equation, where kinetic parameters from previously performed batch experiments were used. Model simulations, obtained by a nonequidistant finite differences numerical solution of a complex equation system, were proved and verified in a set of experiments performed in a microreactor. Based on the developed model, further microreactor design and process optimization are feasible. 

It is extremely common for intermediates to occur after the initial enzyme-substrate complex, as in equation 3.19. However, it is often found for physiological substrates that these intermediates do not accumulate and that the slow step in equation 3.19 is k2. (A theoretical reason for this is discussed in Chapter 12, where examples are given.) Under these conditions, KM is equal to Ks, the dissociation constant, and the original Michaelis-Menten mechanism is obeyed to all intents and purposes. The opposite occurs in many laboratory experiments. The enzyme kineticist often uses synthetic, highly reactive substrates to assay enzymes, and covalent intermediates frequently accumulate. 

To motivate the form of the experiments, note first that two parameters are properties of the organism the m and the a of the chemostat equations. One might postulate that the competitor with the largest m or the one with the smallest a should win the competition. Recall that m is the maximal growth rate and that a (the Michaelis-Menten constant) represents the half-saturation concentration (and so is an indicator of how well an organism thrives at low concentrations). Both of these quantities are obtainable in the laboratory by growing the organism (without a competitor) on the nutrient. (Hansen and Hubbell used a Lineweaver-Burk plot.)... 

The experimental and mathematical models were divided into two hierarchical steps, as seen in Fig. 5. First, hydrolysis experiments were conducted, and the hydrolysis time profile was matched to hydrolysis rate equations. A separate hydrolysis-only model was used to match the hydrolysis data to Michaelis-Menten based kinetics and to solve for unknown parameters. Second, SSF experiments were conducted using identical enzyme loading, and these datasets were matched to a complete SSF model. The SSF model incorporated the hydrolysis parameters from the first step and was used to solve for the unknown fermentation parameters using Monod-based kinetics. 

Deep knowledge of the enzymatic reaction is necessary for a proper selection of the variables that should be considered in the reaction model. In this case, two variables were selected Orange n concentration, as the dye is the substrate to be oxidized, and H2O2 addition rate, as the primary substrate of the enzyme (Lopez et al. 2007). The performance of some discontinuous experiments at different initial values of both variables resulted in the definition of a kinetic equation, defined using a Michaelis-Menten model with respect to the Orange II concentration and a first-order linear... 

The experimental results fit well to the rate equation derived from a Michaelis-Menten model as seen in Figure 2.2. The two-model parameters are easily obtained by fitting the model (Equation 2.31) to the experiments. For Cl = Kjii, the reaction rate corresponds to half of the maximum value. 

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