## The Michaelismenten Hill Equations Model The Effects Of Substrate Concentration

The Michaelis-Menten Equation

The Michaelis-Menten equation (29) illustrates in mathematical terms the relationship between initial reaction velocity v; and substrate concentration [S], shown graphically in Figure 8-3.

The Michaelis constant Km is the substrate concentration at which v is half the maximal velocity (Vmax/2) attainable at a particular concentration of enzyme. Km thus has the dimensions of substrate concentration. The dependence of initial reaction velocity on [S] and Km may be illustrated by evaluating the Michaelis-Menten equation under three conditions.

(1) When [S] is much less than Km (point A in Figures 8-3 and 8-4), the term Km + [S] is essentially equal to Km. Replacing Km + [S] with Km reduces equation (29) to vi —-

Vmax[S]

Vmax[S] Km

where ~ means "approximately equal to." Since Vmax and Km are both constants, their ratio is a constant. In other words, when [S] is considerably below Km, v; ^ k[S]. The initial reaction velocity therefore is directly proportionate to [S].

(2) When [S] is much greater than Km (point C in Figures 8-3 and 8-4), the term Km + [S] is essentially equal to [S]. Replacing Km + [S] with [S] reduces equation (29) to

Thus, when [S] greatly exceeds Km, the reaction velocity is maximal (V^) and unaffected by further increases in substrate concentration.

(3) When [S] = Km (point B in Figures 8-3 and 8-4).

Equation (32) states that when [S] equals Km, the initial velocity is half-maximal. Equation (32) also reveals that Km is—and may be determined experimentally from— the substrate concentration at which the initial velocity is half-maximal.

A Linear Form of the Michaelis-Menten Equation Is Used to Determine Km & Vmax

The direct measurement of the numeric value of Vmax and therefore the calculation of Km often requires im-practically high concentrations of substrate to achieve saturating conditions. A linear form of the MichaelisMenten equation circumvents this difficulty and permits Vmax and Km to be extrapolated from initial velocity data obtained at less than saturating concentrations of substrate. Starting with equation (29), vi —

invert factor vi

and simplify

VVmax J

Equation (35) is the equation for a straight line, y = ax + b, wherey = 1/v; and x = 1/[S]. A plot of 1/v; asy as a function of 1/[S] as x therefore gives a straight line whose y intercept is l/V^ and whose slope is Km/Vmax. Such a plot is called a double reciprocal or Lineweaver-Burk plot (Figure 8-5). Setting the y term of equation (36) equal to zero and solving for x reveals that the x intercept is -1/Km.

Stated another way, the smaller the tendency of the enzyme and its substrate to dissociate, the greater the affinity of the enzyme for its substrate. While the Michaelis constant Km often approximates the dissociation constant Kd, this is by no means always the case. For a typical enzyme-catalyzed reaction,

Km is thus most easily calculated from the x intercept.

Km May Approximate a Binding Constant

The affinity of an enzyme for its substrate is the inverse of the dissociation constant Kd for dissociation of the enzyme substrate complex ES.

Figure 8-5. Double reciprocal or Lineweaver-Burk plot of 1/v, versus 1/[S] used to evaluate Km and Vmax.

Figure 8-5. Double reciprocal or Lineweaver-Burk plot of 1/v, versus 1/[S] used to evaluate Km and Vmax.

Hence, 1/Km only approximates 1/Kd under conditions where the association and dissociation of the ES complex is rapid relative to the rate-limiting step in catalysis. For the many enzyme-catalyzed reactions for which k-1 + k2 is not approximately equal to k-1, 1/Km will underestimate 1/Kd.

The Hill Equation Describes the Behavior of Enzymes That Exhibit Cooperative Binding of Substrate

While most enzymes display the simple saturation kinetics depicted in Figure 8-3 and are adequately described by the Michaelis-Menten expression, some enzymes bind their substrates in a cooperative fashion analogous to the binding of oxygen by hemoglobin (Chapter 6). Cooperative behavior may be encountered for multimeric enzymes that bind substrate at multiple sites. For enzymes that display positive cooperativity in binding substrate, the shape of the curve that relates changes in v; to changes in [S] is sigmoidal (Figure 8-6). Neither the Michaelis-Menten expression nor its derived double-reciprocal plots can be used to evaluate cooperative saturation kinetics. Enzymologists therefore employ a graphic representation of the Hill equation originally derived to describe the cooperative binding of O2 by hemoglobin. Equation (43) represents the Hill equation arranged in a form that predicts a straight line, where k' is a complex constant.

Figure 8-6. Representation of sigmoid substrate saturation kinetics.

Figure 8-6. Representation of sigmoid substrate saturation kinetics.

Equation (43) states that when [S] is low relative to k', the initial reaction velocity increases as the nth power of [S].

A graph of log vi/(Vmax — v;) versus log[S] gives a straight line (Figure 8-7), where the slope of the line n is the Hill coefficient, an empirical parameter whose value is a function of the number, kind, and strength of the interactions of the multiple substrate-binding sites on the enzyme. When n = 1, all binding sites behave independently, and simple Michaelis-Menten kinetic behavior is observed. If n is greater than 1, the enzyme is said to exhibit positive cooperativity. Binding of the

Figure 8-7. A graphic representation of a linear form of the Hill equation is used to evaluate S50, the substrate concentration that produces half-maximal velocity, and the degree of cooperativity n.

Figure 8-7. A graphic representation of a linear form of the Hill equation is used to evaluate S50, the substrate concentration that produces half-maximal velocity, and the degree of cooperativity n.

first substrate molecule then enhances the affinity of the enzyme for binding additional substrate. The greater the value for n, the higher the degree of cooperativity and the more sigmoidal will be the plot of v; versus [S]. A perpendicular dropped from the point where the y term log vi/(Vmax — v;) is zero intersects the x axis at a substrate concentration termed S50, the substrate concentration that results in half-maximal velocity. S50 thus is analogous to the P50 for oxygen binding to hemoglobin (Chapter 6).

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