2 Структура и функция белка (1160071), страница 33
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Substituting this into Equation 8-17 simplifies the expression to[ES] =[Et][S]Km + [SI(8-18)Step 4. Vo can now be expressed in terms of IES]. Equation 8-18 isused to substitute for LES] in Equation 8-10, givingKm + [S](8-19)This equation can be further simplified. Because the maximum velocity will occur when the enzyme is saturated and LES] = [Et], Vmax canbe defined as £2LEt]. Substituting this in Equation 8-19 givesV¥ m a x [Slmaxo -[SJKmV= V[SI(mM)consistent with the plateau observed at high [SI.The Michaelis-Menten equation is therefore consistent with the observed dependence of Vo on [SI,with the shape of the curve defined by the termsVmax/Km at low [SJ and Vmax at high [S].(8-20)This is the Michaelis-Menten equation, the rate equation for a onesubstrate, enzyme-catalyzed reaction.
It is a statement of the quantitative relationship between the initial velocity Vo, the maximum initialvelocity Vmax, and the initial substrate concentration [S], all relatedthrough the Michaelis-Menten constant Km. Does the equation fit thefacts? Yes; we can confirm this by considering the limiting situationswhere [SJ is very high or very low, as shown in Figure 8-12.An important numerical relationship emerges from the Michaelis—Menten equation in the special case when Vo is exactly one-half Vmax(Fig. 8-12).
ThenYvFigure 8—12 Dependence of initial velocity on substrate concentration, showing the kinetic parameters that define the limits of the curve at high andlow [S]. At low [S], Km » [S], and the [S] term inthe denominator of the Michaelis—Menten equation(Eqn 8-20) becomes insignificant; the equation simplifies to Vo = Vmax[S]/Km, and Vo exhibits a lineardependence on [S], as observed.
At high [S], where[S] ^> Km, the Km term in the denominator of theMichaelis-Menten equation becomes insignificant,Km + [S]Ymax2v[SIKm max+ [SJ(8-21)On dividing by Vn , we obtain12[S|1K~ + [J31(8-22)Solving for Km, we get Km + [SJ - 2[S], oro = JV max(8-23)This represents a very useful, practical definition of Km: Km is equivalent to that substrate concentration at which Vo is one-\ia^ V max . "Notethat Km has units of molarity.The Michaelis-Menten equation (8-20) can be algebraically transformed into forms that are useful in the practical determination of Kmand Vmax (Box 8-1) and, as we will describe later, in the analysis ofinhibitor action (see Box 8-2).Chapter 8 EnzymesBOX 8-1215Transformations of the Michaelis—Menten Equation: The Double-Reciprocal PlotThe Michaelis-Menten equation:o -V* maxmax[S]Km + [SIcan be algebraically transformed into forms thatare more useful in plotting experimental data. Onecommon transformation is derived simply by taking the reciprocal of both sides of the MichaelisMenten equation to give1VoKm + [S]Vmax[S]Separating the components of the numerator onthe right side of the equation givesVmax[S][S]Vmax[S]which simplifies tovmvmThis equation is a transform of the MichaelisMenten equation called the Lineweaver-Burkequation.
For enzymes obeying the MichaelisMenten relationship, a plot of 1/VO versus 1/[S] (the"double-reciprocal" of the V0-versus-[S] plot wehave been using to this point) yields a straight line(Fig. 1). This line will have a slope of Km/Vmax, anintercept of UVmax on the Wo axis, and an intercept of -UKm on the 1/[S] axis. The double-reciprocal presentation, also called a Lineweaver-Burkplot, has the great advantage of allowing a moreaccurate determination of Vmax> which can only beapproximated from a simple plot of Vo versus IS](see Fig. 8-12).The Meaning of V max and Km Is Unique for Each EnzymeIt is important to distinguish between the Michaelis-Menten equationand the specific kinetic mechanism upon which it was originally based.The equation describes the kinetic behavior of a great many enzymes,and all enzymes that exhibit a hyperbolic dependence of Vo on [S] aresaid to follow Michaelis-Menten kinetics.
The practical rule thatKm = [S] when Vo = iV max (Eqn 8-23) holds for all enzymes that followMichaelis—Menten kinetics (the major exceptions to Michaelis—Menten kinetics are the regulatory enzymes, discussed at the end ofthis chapter). However, this equation does not depend on the relativelysimple two-step reaction mechanism proposed by Michaelis and Menten (Eqn 8-9). Many enzymes that follow Michaelis-Menten kineticshave quite different reaction mechanisms, and enzymes that catalyzeFigure 1A double-reciprocal, orLineweaver-Burk, plot.Other transformations of the MichaelisMenten equation have been derived and used.Each has some particular advantage in analyzingenzyme kinetic data.The double-reciprocal plot of enzyme reactionrates is very useful in distinguishing between certain types of enzymatic reaction mechanisms (seeFig.
8-14) and in analyzing enzyme inhibition (seeBox 8-2).216Part II Structure and Catalysisreactions with six or eight identifiable steps will often exhibit the samesteady-state kinetic behavior. Even though Equation 8-23 holds truefor many enzymes, both the magnitude and the real meaning of Vmaxand Km can change from one enzyme to the next. This is an importantlimitation of the steady-state approach to enzyme kinetics. Vmax andKm are parameters that can be obtained experimentally for any givenenzyme, but by themselves they provide little information about thenumber, rates, or chemical nature of discrete steps in the reaction.Steady-state kinetics nevertheless represents the standard languageby which the catalytic efficiencies of enzymes are characterized andcompared.
We now turn to the application and interpretation of theterms Vmax and Km.A simple graphical method for obtaining an approximate value forKm is shown in Figure 8-12. A more convenient procedure, using adouble-reciprocal plot, is presented in Box 8-1. The Km can varygreatly from enzyme to enzyme, and even for different substrates ofthe same enzyme (Table 8-6). The term is sometimes used (inappropriately) as an indication of the affinity of an enzyme for its substrate.Table 8-6 Km for some e n z y m e sEnzymeCatalaseHexokinase (brain)SubstrateH2O2ATPD-GlucoseD-FructoseKm (mM)250.40.051.5Carbonic anhydraseChymotrypsinHCO3GlycyltyrosinylglycineAf-Benzoyltyrosinamide/3-GalactosidaseThreonine dehydrataseD-Lactose4.0L-Threonine5.091082.5The actual meaning of Km depends on specific aspects of the reactionmechanism such as the number and relative rates of the individualsteps of the reaction.
Here we will consider reactions with two steps.On page 214 Km is defined by the expressionKm = ^ 4 ^(8-24.For the Michaelis-Menten reaction, k2 is rate-limiting; thus k2 <^ k-iand Km reduces to k-^lki, which is defined as the dissociation constant, Ks, for the ES complex. Where these conditions hold, Km doesrepresent a measure of the affinity of the enzyme for the substrate inthe ES complex.
However, this scenario does not apply to all enzymes.Sometimes k2 ^> k-i, and then Km = k2lk\. In other cases, k2 and k-iare comparable, and Km remains a more complex function of all threerate constants (Eqn 8-24). These situations were first analyzed byHaldane along with George E. Briggs in 1925. The Michaelis-Mentenequation and the characteristic saturation behavior of the enzyme stillapply, but Km cannot be considered a simple measure of substrate affinity. Even more common are cases in which the reaction goes throughmultiple steps after formation of the ES complex; Km can then becomea very complex function of many rate constants.Chapter 8 EnzymesVmax also varies greatly from one enzyme to the next. If an enzymereacts by the two-step Michaelis-Menten mechanism, Vmax is equivalent to &2tEt], where k2 is the rate-limiting step. However, the numberof reaction steps and the identity of the rate-limiting step(s) can varyfrom enzyme to enzyme.
For example, consider the quite common situation where product release, EP -» E + P, is rate-limiting:E+P(8-25)In this case, most of the enzyme is in the EP form at saturation, and7 m a x = &3[EJ. It is useful to define a more general rate constant, kcat,to describe the limiting rate of any enzyme-catalyzed reaction at saturation. If there are several steps in the reaction, and one is clearlyrate-limiting, kcat is equivalent to the rate constant for that limitingstep. For the Michaelis-Menten reaction, kcat = k2. For the reaction ofEquation 8-25, kcat = k3.
When several steps are partially rate-limiting, kcat can become a complex function of several of the rate constantsthat define each individual reaction step. In the Michaelis-Mentenequation, kcat = Vmax/[Et], and Equation 8-19 becomesy* cat [E t ][S](8 26)°=^Ttsf"The constant kcat is a first-order rate constant with units of reciprocaltime, and is also called the turnover number. It is equivalent to thenumber of substrate molecules converted to product in a given unit oftime on a single enzyme molecule when the enzyme is saturated withsubstrate. The turnover numbers of several enzymes are given in Table8-7.Table 8-7 Turnover numbers* (k^) of some enzymesEnzymeSubstrateCatalaseCarbonic anhydraseAcetylcholinesterase/3-LactamaseFumaraseRecA protein (ATPase)H2O2HCO3AcetylcholineBenzylpenicillinFumarateATPk^t (s"1)40,000,000400,00014,0002,0008000.4* Number of substrate molecules transformed per second per molecule ofenzyme.The kinetic parameters kcat and Km are generally useful for thestudy and comparison of different enzymes, whether their reactionmechanisms are simple or complex.
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