Enzyme Kinetics as an Approach to Understanding Mechanism:- The Relationship between Substrate Concentration and Reaction Rate Can Be Expressed Quantitatively

The curve expressing the relationship between [S] and V0 (Fig. 6–11) has the same general shape for most enzymes (it approaches a rectangular hyperbola), which can be expressed algebraically by the Michaelis-Menten equation. Michaelis and Menten derived this equation starting from their basic hypothesis that the rate limiting step in enzymatic reactions is the breakdown of the ES complex to product and free enzyme. The equation is

The important terms are [S], V0, Vmax, and a constant called the Michaelis constant, Km. All these terms are readily measured experimentally.

Here we develop the basic logic and the algebraic steps in a modern derivation of the Michaelis-Menten equation, which includes the steady-state assumption introduced by Briggs and Haldane. The derivation starts with the two basic steps of the formation and break down of ES (Eqns 6–7 and 6–8). Early in the reaction, the concentration of the product, [P], is negligible, and we make the simplifying assumption that the reverse re action, P→S (described by k_-2), can be ignored. This assumption is not critical but it simplifies our task. The overall reaction then reduces to

V0is determined by the breakdown of ES to form product, which is determined by [ES]:

V₀=k₂[ES] (6–11)

Because [ES] in Equation 6–11 is not easily measured experimentally, we must begin by finding an alternative expression for this term. First, we introduce the term [Et], representing the total enzyme concentration (the sum of free and substrate-bound enzyme). Free or un bound enzyme can then be represented by [Et] [ES]. Also, because [S] is ordinarily far greater than [Et], the amount of substrate bound by the enzyme at any given time is negligible compared with the total [S]. With these conditions in mind, the following steps lead us to an ex pression for V0in terms of easily measurable parameters. Step 1 The rates of formation and breakdown of ES are determined by the steps governed by the rate constants k1 (formation) and k_-1+k₂ (breakdown), according to the expressions

Rate of ES formation=k₁([Et]-[ES]) [S] (6–12)

Rate of ES breakdown= k_-1[ES]+k2[ES] (6–13)

Step 2 We now make an important assumption: that the initial rate of reaction reflects a steady state in which [ES] is constant—that is, the rate of formation of ES is equal to the rate of its breakdown. This is called the steady-state assumption. The expressions in Equa tions 6–12 and 6–13 can be equated for the steady state, giving k1([Et]-[ES]) [S]=k_-1[ES]+k₂[ES] (6–14) Step 3 In a series of algebraic steps, we now solve Equation 6–14 for [ES]. First, the left side is multiplied out and the right side simplified to give k1[Et][S]-k1[ES][S]=(k_-1+k₂) [ES] (6–15) Adding the term k1[ES][S] to both sides of the equation and simplifying gives k1[Et][S]=(k₁[S] k_-1+k₂) [ES] (6–16) We then solve this equation for [ES]:

This can now be simplified further, combining the rate constants into one expression:

The term (k₂+k-₁)/k₁ is defined as the Michaelis constant, Km. Substituting this into Equation 6–18 simplifies the expression to

Step 4 We can now express V0 in terms of [ES]. Substituting the right side of Equation 6–19 for [ES] in Equation 6–11 gives

This equation can be further simplified. Because the maximum velocity occurs when the enzyme is saturated (that is, with [ES]=[Et]) V_max can be defined as k2[Et]. Substituting this in Equation 6–20 gives Equation 6–9:

This is the Michaelis-Menten equation, the rate equation or a one-substrate enzyme-catalyzed reaction. It is a statement of the quantitative relationship between the initial velocity V0, the maximum velocity Vmax, 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. An important numerical relationship emerges from the Michaelis-Menten equation in the special case when V0 is exactly one-half V_max (Fig. 6–12). Then

On dividing by Vmax, we obtain

Solving for Km, we get Km [S] 2[S], or

This is a very useful, practical definition of K_m: Km is equivalent to the substrate concentration at which V₀ is one-half V_max.

The Michaelis-Menten equation (Eqn 6–9) can be algebraically transformed into versions that are useful in the practical determination of Km and Vmax (Box 6–1) and, as we describe later, in the analysis of inhibitor action.

FIGURE 6–12 Dependence of initial velocity on substrate concentration. This graph shows the kinetic parameters that define the limits of the curve at high and low [S]. At low [S], Km>>[S] and the [S] term in the denominator of the Michaelis-Menten equation (Eqn 6–9) becomes insignificant. The equation simplifies to V₀=V_max[S]/Km and V0 exhibits a linear dependence on [S], as observed here. At high [S], where [S]>>K_m, the Km term in the denominator of the Michaelis Menten equation becomes insignificant and the equation simplifies to V₀=V_max; this is consistent with the plateau observed at high [S]. The Michaelis-Menten equation is therefore consistent with the observed dependence of V0 on [S], and the shape of the curve is defined by the terms V_max/K_m at low [S] and V_max at high [S].

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