Maximal velocity determination

Saturation kinetics are also called zero-order kinetics or Michaelis-Menten kinetics. The Michaelis-Menten equation is mainly used to characterize the interactions of enzymes and substrates, but it is also widely applied to characterize the elimination of chemical compounds from the body. The substrate concentration that produces half-maximal velocity of an enzymatic reaction, termed value or Michaelis constant, can be determined experimentally by graphing r/, as a function of substrate concentration, [S]. 

Figure 22 Examples of enzyme kinetic plots used for determination of Km and Vmax for a normal and an allosteric enzyme Direct plot [(substrate) vs. initial rate of product formation] and various transformations of the direct plot (i.e., Eadie-Hofstee, Lineweaver-Burk, and/or Hill plots) are depicted for an enzyme exhibiting traditional Michaelis-Menten kinetics (coumarin 7-hydroxylation by CYP2A6) and one exhibiting allosteric substrate activation (testosterone 6(3-hydroxylation by CYP3A4/5). The latter exhibits an S-shaped direct plot and a hook -shaped Eadie-Hofstee plot such plots are frequently observed with CYP3A4 substrates. Km and Vmax are Michaelis-Menten kinetic constants for enzymes. K is a constant that incorporates the interaction with the two (or more) binding sites but that is not equal to the substrate concentration that results in half-maximal velocity, and the symbol n (the Hill coefficient) theoretically refers to the number of binding sites. See the sec. III.C.3 for additional details.

Vmax is the velocity of an enzyme-catalyzed reaction when the enzyme is saturated with all of its substrates and is equal to the product of the rate constant for the rate-limiting step of the reaction at substrate saturation (kCiU) times the total enzyme concentration, Ex, expressed as molar concentration of enzyme active sites. For the very simple enzyme reaction involving only one substrate described by Equation II-4, kCM = . Elowever, more realistic enzyme reactions involving two or more substrates, such as described by Equations II-11 and 11-12, require several elementary rate constants to describe their mechanisms. It is not usually possible to determine by steady-state kinetic analysis which elementary rate constant corresponds to kcat. Nonetheless, it is common to calculate kcat values for enzymes by dividing the experimentally determined Fmax, expressed in units of moles per liter of product formed per minute (or second), by the molar concentration of the enzyme active sites at which the maximal velocity was determined. The units of cat are reciprocal time (min -1 or sec - x) and the reciprocal of cat is the time required for one enzyme-catalyzed reaction to occur. kcat is also sometimes called the turnover number of the enzyme. 

In a study of the highly purified alanine racemase of E. coli, Lambert and Neuhaus determined significant differences in the maximal velocities and the Michaelis-Menten constants of the substrates in the forward (L - dl) and reverse directions (d - dl) [37]. From these data the value calculated for Keq is 1.11 0.15. The time course of the reaction showed that in 10 min with L-alanine as substrate ca. 0.09 jumol of D-alanine were formed. With the same amount of enzyme (750 ng) and in the same time period, ca. 0.05 jamol of L-alanine were formed from D-alanine. Similar results have been reported for the same enzyme from S. faecalis and for proline racemases [37]. Thus, in these cases, there are definite kinetic differences, as expected for the existence of two diastereoisomers formed between enzyme and two substrate enantiomers. 

If we set up the same enzyme assay with a fixed amount of enzyme but vary the substrate concentration we will observe that initial velocity (va) will steadily increase as we increase substrate concentration ([S]) but at very high [S] the va will asymptote towards a maximal value referred to as the Vmax (or maximal velocity). A plot of va versus [S] will yield a hyperbola, that is, v0 will increase until it approaches a maximal value. The initial velocity va is directly proportional to the amount of enzyme—substrate complex (E—S) and accordingly when all the available enzyme (total enzyme or E j) has substrate bound (i.e. E—S = E i -S and the enzyme is completely saturated ) we will observe a maximal initial velocity (Pmax)- The substrate concentration for half-maximal velocity (i.e. the [S] when v0 = Vmax/2) is termed the Km (or the Michaelis—Menten constant). However because va merely asymptotes towards fT ax as we increase [S] it is difficult to accurately determine Vmax or Am by this graphical method. However such accurate determinations can be made based on the Michaelis-Menten equation that describes the relationship between v() and [S],... 

The MTT assay can also be used to quantify cell activation by determining the maximal velocity (F) of the reaction (Gerlier Thomasset, 1986). Alternatively, in order to compare two states of activation of a given cell, the MTT formazan produced by the same number of viable cells can be measured. 

Adjacent to the mucosa, no agitation is possible and the transport rate of most drugs is determined by passive diffusion in the aqueous medium. The thickness of the layer in situ amounts to about 500 pm and in vitro to about 150-200 pm. The average unstirred layer thickness can be calculated from Michaelis-Menten constant and maximal velocity of maltose hydrolysis by intestinal maltase. In... 

Vm a is the maximal velocity for the reaction, and Km is a rate constant. Experimental determinations of Km and Vlna, are made with Lineweaver-Burk double-reciprocal plots. 

A EXPERIMENTAL FIGURE 3-19 The and l/ ,ax for an enzyme-catalyzed reaction are determined from plots of the initial velocity versus substrate concentration. The shape of these hypothetical kinetic curves is characteristic of a simple enzyme-catalyzed reaction in which one substrate (S) is converted into product (P). The initial velocity is measured immediately after addition of enzyme to substrate before the substrate concentration changes appreciably, (a) Plots of the initial velocity at two different concentrations of enzyme [E] as a function of substrate concentration [S]. The [S] that yields a half-maximal reaction rate is the Michaelis constant K, a measure of the affinity of E for S. Doubling the enzyme concentration causes a proportional increase in the reaction rate, and so the maximal velocity 1/max is doubled the K, however, is unaltered, (b) Plots of the initial velocity versus substrate concentration with a substrate S for which the enzyme has a high affinity and with a substrate S for which the enzyme has a low affinity. Note that the 1/max is the same with both substrates but that is higher for S, the low-affinity substrate. 

From plots of reaction rate versus substrate concentration, two characteristic parameters of an enzyme can be determined the Michaelis constant K, a measure of the enzyme s affinity for substrate, and the maximal velocity V ax> a measure of its catalytic power (see Figure 3-19). 

Here, the first term represents the resistance due to eddy transport, and the second term that of molecular diffusion. The term B = 0.1 m/s depends on the type of surface, but it is only mildly dependent on u (Chamberlain, 1966 Garland, 1977). In this manner, rg(z) becomes amenable to calculation, and Table 1-11 presents some numerical values that were given by Garland (1979). The results are maximal velocities, as they do not yet include the effect of the surface resistance rs. For the uptake of gases by soils and vegetation, rs must be considered an empirical quantity to be determined by experiments. 

The enzyme catalyzes the incorporation of one atom of oxygen into L-lysine, and 8-aminonorvaleramide is formed concomitantly with the evolution of carbon dioxide. Enzyme activity was estimated by the polarographic determination of oxygen consumption. The concentration curve of L-lysine was sigmoidal, and 0.18 mM L-lysine was required for the half maximal velocity of the enzyme. 

The enzymatic activity of AChE from electric eel is inhibited by monosulfonate tetraphenyl porphyrin (TPPSi) the structure of which can be seen in Figure 12.1 with Rj = SO3, R2 = no substituent group, and no metal incorporated [36]. A Lineweaver-Burk plot of enzymatic rates in the absence/presence of TPPSj determines the type of inhibition resulting from the presence of the porphyrin. The Lineweaver-Burk plot is the plot of the double-reciprocal form of the initial enzymatic rate versus the substrate concentration. Intersection of the lines generated in the absence and presence of inhibitor at the y-axis shows no change in maximal velocity but a change in the Michaelis constant K j, an indication of the substrate binding affinity, and indicates competitive inhibition by the porphyrin. Competitive inhibition involves competition of the inhibitor for occupation of the active site of the enzyme. 

The Michaelis constant is obviously inversely proportional to the affinity of the enzyme for the substrate, and is numerically equal to the substrate concentration when the reaction has reached half its maximal velocity. The dimensions of K and are recorded as g mol per litre. However, they are not true equilibrium constants but ratios of velocity constant for the forward and reverse reactions. A suitable determination of is by the graphical method of Lineweaver and Burk (1934), where the initial rate of formation of ES is plotted against substrate concentration, both as reciprocals.This should give a straight line and, if it does, the value of is shown at the intersection of slope and abscissa. 

Another frequently used method for pK determination is based on measurements of the pH dependence of enzyme kinetics. The pH influences the kinetics of the enzymatic reaction in three ways. It affects maximal velocity, the formation of the enzyme-substrate complex, and the stability of the protein. 

Studies of enzyme kinetics involve measurements of initial velocity v of the reaction as a function of substrate concentration [S]. Values of the kinetic constants are determined by fitting the initial velocity and concentration data to the appropriate rate equations by the least-squares method. The maximal velocity and Michaelis constant are obtained from experimental data, usually from different methods of plotting the kinetic parameters, the most common of which is to plot llv versus... 

Km is known as the Michaelis constant and has. in neral. no physical meaning. Its numerical value is that of the substrate concentration at half-maximal velocity it is expressed in units of molarity (M). Figure 8 illustrates the graphical determination of Km for a reaction measured in the presence of enzyme. At that concentration, half the enzyme molecules are bound to the substrate (cf. central part of Fig. 9). 

The Michaelis-Menten equation (32) is a rectangular hyperbola which can only be drawn accurately with a large number of experimental points. The Michaelis constant Km), although situated in an easily accessible part of the curve, can only be calculated after the maximal velocity has been determined by extrapolation to infinite substrate concentration. 

Estimations of enzyme kinetic constants have been used previously to explain the substrate selectivity exhibited by pea and spinach stromal ATI (Frentzen, eta/., 1983). We have determined the acyl substrate affinity constants and maximal velocities for the enzymes isolated from chloroplasts of three chilling sensitive species and will rely upon previously published values for spinach and pea for our discussion. We have not determined Km values for G3P in tne presence of acyl-ACPs for any of these enzymes. 

The value of Np required to achieve a desired resolution is determined by Eq. (16-168) or (16-171). Since N = L/HTU 2Np = 2L/HETP, Fig. 16-13 or Eq. (16-183) can be used to determine the range of the dimensionless velocity ReSc that maximizes Np for a given particle diameter and column length. 

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