Half-maximal velocity

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]. 

The allosteric model just presented is called a K system because the concentration of substrate giving half-maximal velocity, defined as -K0 5, changes in response to effectors (Figure 15.11). Note that Vjnax is constant in this system. 

Figure 41-11. A comparison of the kinetics of carrier-mediated (facilitated) diffusion with passive diffusion. The rate of movement in the latter is directly proportionate to solute concentration, whereas the process is saturable when carriers are involved. The concentration at half-maximal velocity is equal to the binding constant (KJ of the carrier for the solute. maximal rate.)...

The affinity of an enzyme for its substrate is usually considered to be the reciprocal of Km, the Michaelis constant, but it is, in fact, the reciprocal of K the dissociation constant of the enzyme-substrate complex. It is seldom that K, can be measured directly, but Km normally provides a good approximation. The latter constant can be found experimentally it is the substrate concentration at which the enzyme displays half-maximal velocity. In the reaction,... 

Allosteric enzymes show relationships between V0 and [S] that differ from Michaelis-Menten kinetics. They do exhibit saturation with the substrate when [S] is sufficiently high, but for some allosteric enzymes, plots of V0 versus [S] (Fig. 6-29) produce a sigmoid saturation curve, rather than the hyperbolic curve typical of non-regulatory enzymes. On the sigmoid saturation curve we can find a value of [S] at which V0 is half-maximal, but we cannot refer to it with the designation Km, because the enzyme does not follow the hyperbolic Michaelis-Menten relationship. Instead, the symbol [S]0 e or K0,5 is often used to represent the substrate concentration giving half-maximal velocity of the reaction catalyzed by an allosteric enzyme (Fig. 6-29). 

A. Vmax is altered. B. The substrate concentration that gives half-maximal velocity (Ko.5) is altered. 

Prove that the Km equals the substrate concentration at one-half maximal velocity. 

ADH CD 3D HEThDP Mes nh PAC PDC PDCS.c. PDCS.u. PDCZ.w. So.5 ThDP v/S v max wt alcohol dehydrogenase circular dichroism three-dimensional 2-(hydroxyethyl)thiamine diphosphate 4-morpholineethanesulfonsaure Hill-coefficient phenylacetyl carbinol pyruvate decarboxylase PDC from Saccharomyces cerevisiae PDC from Saccharomyces uvarum PDC from Zymomonas mobilis substrate concentration necessary for half-maximal velocity thiamine diphosphate velocity vs substrate concentration maximal velocity wild-type... 

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.

Equation (19.7) assumes that the system is at equilibrium. To make sense of it, think about a few different values for [ligand]. When [ligand] = 0, the fractional occupancy equals zero. When [ligand] is very, very high (i.e., many times the KD), the fractional occupancy approaches 100%. When [ligand] = KD, fractional occupancy is 50% (just as the Michaelis-Menten constant Km describes the concentration of enzyme substrate that gives half-maximal velocity). 

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],... 

Note that when = [S], u = 7/2, i.e. K = substrate concentration for half maximal velocity.)... 

Ks is the dissociation constant for the enzyme substrate complex. It is important to remember that the Michaelis-Menten equation holds true not only for the mechanism as stated above, but for many different mechanisms that are not included in this treatment. In summary, ITm can be described as an apparent dissociation constant for all enzyme-bound species and, in all cases, it is the substrate concentration at which the enzyme operates at half-maximal velocity. 

Hence, Michaelis - Menten constant, Km, is defined as the substrate concentration at half maximal velocity and is expressed as mole per litre (Fig. 6.2). 

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 ... 

The constant K in the above equation no longer equals the substrate concentration that yields half-maximal velocity (except when n = 1, when the equation reduces to the Henri-Michaelis-Menten equation). 

Relationship between the initial velocity (v) and the substrate concentration [S] for an allosteric enzyme that shows a homotropic effect. The substrate functions as a positive modulator. The profile is sigmoidal, and during the steep part of the profile, small changes in [S] can cause large changes in v. Ko.i represents the substrate concentration corresponding to half-maximal velocity. 

The value of the substrate concentration corresponding to half-maximal velocity is designated as A o.5 and not since the allosteric kinetics do not follow the hyperbolic Michaelis-Menten relationship. 

Hence in studying enzyme catalysed reactions, one rarely looks at maximum velocity, but tries instead to measure the initial velocity of the reaction, i.e. the reaction rate over the first part of the graph of Figure 12. If one looks at a substrate concentration that gives half-maximal velocity - Vm - (shown in Figure 33), this concentration turns out to be a constant for any... 

Km is the substrate concentration at which the enzyme has half-maximal velocity. 

The catalytic action of an enzyme on a given substrate can be described by two parameters 4iax the maximal velocity of the reaction at saturating substrate concentrations, and (the Michaelis constant), a measure of the affinity of an enzyme for its substrate (Figure 3-19). The is defined as the substrate concentration that yields a half-maximal reaction rate (l.e., 5 Knax)- The smaller the value of K, the more avidly an enzyme can bind substrate from a dilute solution and the smaller the substrate concentration needed to reach half-maximal velocity. 

This prototype GK activator enhanced the catalytic activity of GK by increasing the maximum velocity (Fmax) and decreasing the glucose concentration at 0.5 Fmax or half-maximal velocity (S0.5), consistent with a mixed-type non-essential activator. In an in vivo setting, oral administration of... 

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