Substituting equation 27 into equation 26 yields the Michaelis-Menten e [S] max [S]+ KM (28) This equation accounts for the kinetic data given in Figure 8-15.At very low substrate concentration, when [S] is much less than KM, V [s]Vmax/KMi that is, the rate is directly proportional to the substrate concentration. At high substrate concentration, when [S] is much greater than KM, V=Vmax; that is, the rate is maximal, independent of substrate concentration. When s> Kmg V=V
When [S] > Km, V = Vmax
The meaning of KM is evident from equation 28. When [S]= KM,then V= Vmax/2. Thus, KM is equal to the substrate concentration at which the reac- tion rate is half its maximal value [S] max [S]+ KM (28)
192Part‖l PROTEINS nax max max 2 c K Substrate concentration [S] Figure 8-15 A plot of the reaction velocity V as a function of the substrate concentra- tion [S] for an enzyme that obeys Michaelis-Menten kinetics(Vmax is the maximal velocity and Km is the Michaelis constant)
Vmax and Km cAn BE DETERMINED BY VARYING THE SUBSTRATE CONCENTRATION The Michaelis constant, KM, and the maximal rate, Vmax, can be readily derived from rates of catalysis measured at different substrate concentra- tions if an enzyme operates according to the simple scheme given in equation 14. It is convenient to transform the Michaclis-Menten equation into one that gives a straight-line plot. This can be done by taking the reciprocal of both sides of equation 28 to give KM 99 max max S The Lineweaver-Burk plot [S] V= V max [S]+ KM (28)
The Lineweaver-Burk plot
A plot of l/Versus 1/[S], called a Lineweaver- Burk plot, yields a straight line with an intercept of 1/Vmax and a slope of KM/Vmax(Figure 8-16) Alternatively, KM and Vmax can be obtained by fitting the data to equa- tion 28 using a computer program Part Il A plot of 1/Versus 1/[S, called a Lineweaver- Burk plot, yields a straight PROTEINS line with an intercept of l/Vmax and a slope of KM/Vmax( Figure 8-16) Alternatively, KM and Vmax can be obtained by fitting the data to equa tion 28 using a computer program Figure 8-16 double-reciprocal plot of enzyme kinetics: 1/v is plotted as a function of 1/[S]. The slope is KM/Vmax Intercept =-1/KM Intercept =1/vna intercept on the vertical axis is 1/Vmax, and the intercept on the horizontal axis is -1/KM
Il. The Eadie- Hofstee Equation One transformation of the Michaelis-Menten equation is the Lineweaver Burk, or double-reciprocal, equation. Multiplying both sides of the Lineweaver-Burk equation by Vm and re- arranging gives the Eadie-Hofstee equation: Vo=(k)o+ v A plot of Vo vs Vo/S for an enzyme-catalyzed reaction is shown below V K V V The blue curve was obtained in the absence of in hibitor. Which of the other curves(A, B, or C)shows the enzyme activity when a competitive inhibitor is added to the reaction mixture?
SIGNIFICANCE OF KM AND Vmax VALUES The KM values of enzymes range widely (Table 8-2). For most enzymes, KM lies between 10-1 and 10-7M. The KM value for an enzyme depends on the particular substrate and also on environmental conditions such as pH, temperature, and ionic strength
Table 8-2 K M values of some enzymes Enzyme Substrate KLum) Chymotrypsin Acetyl-L-tryptophanamide 5000 Lysozyme Hexa-N-acetylglucosamine 6 B-Galactosidase Lactose 4000 Threonine deaminase Threonine 5000 Carbonic anhydrase CO 8000 Penicillinase Benzylpenicillin 50 ruvate carboxylase Pyruvate 400 HCO3 1000 ATP Arginine-tRNA synthetase Arginine tRNA 0.4 ATP 300
The Michaelis constant km has two meanings.First,KM is the concentration of substrate at which half the active sites are filled. Once the Km is known, the fraction of sites filled Es at any substrate concentration can be calculated from / V [S] ES (30 max [S]+ Km
Second, Km is related to the rate constants of the individual steps in the catalytic scheme given in equation 14. In equation 20, KM is defined as (k2+ k3)/kl. Consider a limiting case in which k2 is much greater than ks. This means that dissociation of the ES complex to E and S is much more rapid than formation of E and product. Under these conditions (k2>k3) E+S ES E+P (14) 2 Equation 19 can be simplified by defining a new constant, KM, called the Michaelis constant k+ k KI M (20)