Michaelis constants, relationships

Figure 3. Schematic view of the substrate uptake rate versus concentration relationship as described by the whole-cell Michaelis-Menten kinetics. Q is the substrate uptake rate, <2max the biologically determined maximum uptake rate per biomass, c the substrate concentration, and Kj the whole-cell Michaelis constant, i.e. the concentration resulting in 2max/2 (mass of substrate per volume). At c

In the case of carboxylesterase-catalyzed hydrolysis (Table 8.1), the Michaelis constant consistently indicated relatively low affinity for the enzyme, with a variation between substrates of one order of magnitude. Even less variation was seen in the maximal velocity of the reaction. It is interesting to note that, for the four compounds where comparisons are possible, a direct relationship exists between the rate of hydrolysis in plasma and the Vmax of carboxylesterase hydrolysis, suggesting comparable catalytic mechanisms. 

Kinetic Haldane relations use a ratio of apparent rate constants in the forward and reverse directions, if the substrate concentrations are very low. For an ordered Bi Bi reaction, the apparent rate constant for the second step is Emax,f/ b (where K, is the Michaelis constant for B) and, in the reverse reaction, V ax,v/Kp. Each of these is multiplied by the reciprocal of the dissociation constant of A and Q, respectively. The forward product is then divided by the reverse product. Hence, the kinetic Haldane relationship for the ordered Bi Bi reaction is Keq = KiO V eJKp)l Kiq V eJKp) = y ,ax.f pKiq/ (yranx,rKmKif). For Completely random mechanisms, thermodynamic and kinetic Haldane relationships are equivalent. 

In Scheme 1, the rate parameters Vmax,f and Emax,r are the maximum velocities in the forward and reverse direction, respectively (such that Emax,f = [Etotai] and Emax,r = ki [Etotai]), a is the Michaelis constant for substrate A (Xa = (/c2 + ksykb), and Xeq is the equihbrium constant (equal to kikslk2k, and having the Haldane relationships... 

Reactions in which the velocity (v) of the process is independent of the reactant concentration, following the rate law v = k. Thus, the rate constant k has units of M sAn example of a zero-order reaction is a Michaelis-Menten enzyme-catalyzed reaction in which the substrate concentration is much larger than the Michaelis constant. Under these conditions, if the substrate concentration is raised even further, no change in the velocity will be observed (since v = Umax)- Thus, the reaction is zero-order with respect to the substrate. However, the reaction is still first-order with respect to total enzyme concentration. When the substrate concentration is not saturating then the reaction ceases to be zero order with respect to substrate. Reactions that are zero-order in each reactant are exceedingly rare. Thus, zero-order reactions address a fundamental difference between order and molecularity. Reaction order is an empirical relationship. Hence, the term pseudo-zero order is actually redundant. All zero-order reactions cease being so when no single reactant is in excess concentration with respect to other reactants in the system. 

Two characteristics, the Michaelis constant KM and the maximal velocity V are the most important numeric data. The well-known Michaelis-Menten equation describes the relationship between the initial reaction rate and the substrate concentration with these two constants. The actual form of the rate equation of an enzymic process depends on the chemical mechanism of the enzymic transformation of the substrate to product (Table 8.1). 

This is the Michaelis-Menten equation, the rate equation for a one-substrate enzyme-catalyzed reaction. It is a statement of the quantitative relationship between the initial velocity V0, the maximum velocity Vnmx, and the initial substrate concentration [S], all related through the Michaelis constant Km. Note that Km has units of concentration. Does the equation fit experimental observations Yes we can confirm this by considering the limiting situations where [S] is very high or very low, as shown in Figure 6-12. 

Tlie kinetic parameters of Eq. 9-44 are Vf, the maximum velocity in the forward direction, the two Michaelis constants, KmB and KmA, and the equilibrium constant Ke, for reversible dissociation of the complex EA and which is equal to k2/k1. The relationship between the parameters of Eq. 9-44 (Km s, V s, and KeqA s) and the rate constants /q- kw is not obvious. However, remember that the parameters are experimental quantities determined by measurements on the enzyme. Sometimes, but not always, it is possible to deduce some of the values of individual rate constants from the experimental parameters. 

Figure C1.1.1 Relationship between substrate concentration and initial velocity at fixed enzyme concentrations. The Michaelis constant, KM, is equal to the substrate concentration corresponding to one-half l/max.

The graphical significance of the constants in the Monod equation are identical to the corresponding constants in the Michaelis-Menten relationship for enzyme kinetics (see Section 5.4.4). The specific growth rate initially increases with increas-... 

Magnetic moment, 153, 155, 160 Magnetic quantum number, 153 Magnetization, 160 Magnetogyric ratio, 153, 160 Main reaction, 237 Marcus equation, 227, 238, 314 Marcus plot, slope of, 227, 354 Marcus theory, applicability of, 358 reactivity-selectivity principle and, 375 Mass, reduced, 189, 294 Mass action law, 11, 60, 125, 428 Mass balance relationships, 19, 21, 34, 60, 64, 67, 89, 103, 140, 147 Maximum velocity, enzyme-catalyzed, 103 Mean, harmonic, 370 Mechanism classification of. 8 definition of, 3 study of, 6, 115 Medium effects, 385, 418, 420 physical theories of, 405 Meisenheimer eomplex, 129 Menschutkin reaction, 404, 407, 422 Mesomerism, 323 Method of residuals, 73 Michaelis constant, 103 Michaelis—Menten equation, 103 Microscopic reversibility, 125... 

Thus, Kn, the Michaelis constant, is a dynamic or pseudo-equilibrium constant expressing the relationship between the actual steady-state concentrations, rather than the equilibrium.concentrations. If Aj, is very small compared to A-i, reduces to K. A steady-state treatment of the more realistic reaction sequence E+ S ES EP E + P yields the same final velocity equation although now Km is a more complex function, composed of the rate constants of all the steps. Thus, the physical significance of K cannot be stated with any certainty in the absence of other data concerning the relative magnitudes of the various rate constants. Nevertheless, represents a valuable constant that relates the velocity of an enzyme-catalyzed reaction to the substrate concentration. Inspection of the Henri-Michaelis-Menten equation shows that Km is equivalent to the substrate concentration that yields half-maximal velocity ... 

Hase (H16) studied the effect of pH on the hydrolysis of acetylcholine by horse serum cholinesterase, and his results have been reanalyzed by Laidler (L5) and extensively discussed by Dixon and Webb (D21). The relationship between pH and the rate of hydrolysis of acetylcholine has been used to obtain information on the structure of the active site of the enzyme (B19, W28). Acetylcholine is a particularly suitable substrate for these studies since it does not change its charge in the pH range studied. Similar pH-activity curves have been obtained using other substrates for cholinesterase (H23, S20, P19). Moreover the pH dependence of enzymic activity varies with the buffer system (K3). By investigating the effect of pH and sodium chloride concentration on the rate of hydrolysis of ben-zoylcholine by human plasma cholinesterase, Kalow (K6) deduced that for this substrate, each enzyme molecule contains at least two binding sites which differ in their dependence on pH. Michaelis constants and maximum hydrolysis velocities were measured for each of the two binding sites, and pK values of the enzyme-substrate complexes were found to be 5.2, 6.7, and 9.2 for one site, and 5.2, 7.0, 8.4, and 8.8 for the other. 

In the 1-substrate case one can at least say that the dissociation constant may not exceed K -, there is therefore a formal mathematical relationship between the two constants. For some mechanisms involving more than one reactant, not even this limited degree of linkage exists. Michaelis constants are empirical kinetic parameters. They have an entirely adequate definition in kinetic terms and should not be equated with thermodynamic constants without sound theoretical or experimental justification. 

This relationship is illustrated in Figure 10.1, The Michaelis constant thus can be determined from a plot of v against [S], by finding the concentration of substrate that gives one-half of the limiting rate. However, this procedure does not provide a very reliable value. 

For enzymes that obey the Michaelis-Menten relationship, a plot of Mr versus 1/[S] the so-called double-reciprocal plot yields a straight line, from which one can obtain more accurate values for the Michaelis constant, Km, and the maximum velocity, Vmax (Figure 6.4). 

The Michaelis constant (Ka), dimension [concentration]. One can easily appreciate from Rg. 4 that the value of the Michaelis constant is obtained when Vo = Vmai/2. Ka is equal to dissociation constant of the enzyme-substrate complex only in the simple Michaelis-Menten mechanism, and in all other cases, it is a complex function of several rate constants (Table 1). However, Ka maybe treated for some purposes as an apparcnr dissociation constant for example, the concentration of free enzyme in solution may be calculated from the relationship ... 

The Michaelis constants and inhibition constants are now rather complex relationships ofMichaehs pH functions. However, the relationships expressed by Eqs. (14.42)-(14.45) are each a function of a single Michaelis pH function (Laidler, 1955 Schulz, 1994). Thus, in this particular mechanism, by a pmdent choice of parameters, one can calculate the pXa values of all forms of the enzyme in reaction (14.34). For example, Eq. (14.42) can be expanded to obtain... 

Michaelis constant can be easily interpreted as the concentration of substrate where the rate of the reaction is half of its maximum. This is, at this concentration, half of the enzyme molecules bind the substrate and, thus, it should be independent of the enzyme concentration. However, this is a rather simplistic conclusion because it considers that all enzymes strictly follow the Michaeiis-Menten behavior over a wide range of experimental conditions. If not, the fCm is likely to equal a much more complex relationship between the different rate constants involved. The clear dependence between Vmax and enzyme concentration enables a direct way for the estimation of enzyme concentration (see Figure 5B). This is the case of many clinical determinations targeting key enzymes for diagnostic purposes or the basis of many other assays, e.g., immunoassays, where enzymes are used as reporters or tracers for easy and sensitive transduction of noncatalytic interactions. 

In the treatments discussed so far, it has been assumed that the back reaction could be neglected. The reactions catalysed by many enzymes are essentially irreversible or the products are immediately subject to further reaction, so that the assumption of irreversibility is valid. However, if the reaction is reversible, the Michaelis equation must be modified. Haldane suggested a notation in which V, and V, are the maximal velocities in the forward and reverse directions, and and K ,p are the Michaelis constants for the substrate and product. The Haldane relationship for a system with a single substrate and single product is then = V,K pA, K s. 

The availability of chemically well defined octamers (A and C) as sDNA substitutes provi s a kinetic tool for the determination of the effects of enzyme inhibitors on the DNA binding site of the enzyme. The 6-amino derivative of 1,2-benzoypyrone, (6-aminocoumarin), competitively inhibits at the octamer duplex A or C sites with an apparent Ki of 28 jiM. Without the inhibition at a fixed NAD+ concentration, a Michaelis-Menten relationship exists between Vinit and the concentration of the octamers (lowest curve in Fig. 3) with an apparent binding constant of 1 jiM. The results shown in Fig. 3 identify a novel site of inhibitors, structurally unrelated to NAD" (12), which act at the DNA binding site of the enzyme. 

The Haldane relationship between the equilibrium constant, maximum velocities, and Michaelis constants.)... 

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