Harpers

ENZYMES: KINETICS / 65= S= EA B CFigure 8–4. Representation of an enzyme at low (A), at high (C), and at a substrate concentrationequal to K m (B). Points A, B, and C correspond to those points in Figure 8–3.THE MICHAELIS-MENTEN & HILLEQUATIONS MODEL THE EFFECTSOF SUBSTRATE CONCENTRATIONThe Michaelis-Menten EquationThe Michaelis-Menten equation (29) illustrates inmathematical terms the relationship between initial reactionvelocity v i and substrate concentration [S],shown graphically in Figure 8–3.V Svi = max [ ]Km+ [ S](29)The Michaelis constant K m is the substrate concentrationat which v i is half the maximal velocity(V max /2) attainable at a particular concentration ofenzyme. K m thus has the dimensions of substrate concentration.The dependence of initial reaction velocityon [S] and K m may be illustrated by evaluating theMichaelis-Menten equation under three conditions.(1) When [S] is much less than K m (point A in Figures8–3 and 8–4), the term K m + [S] is essentially equalto K m . Replacing K m + [S] with K m reduces equation(29) toVVVv1=v1≈ ≈ ⎛ ⎞max[ S]max[ S]K +K ⎝ ⎜ max⎟ [ S]m [ S]m Km⎠(30)where ≈ means “approximately equal to.” Since V maxand K m are both constants, their ratio is a constant. Inother words, when [S] is considerably below K m , v i ∝k[S]. The initial reaction velocity therefore is directlyproportionate to [S].(2) When [S] is much greater than K m (point C inFigures 8–3 and 8–4), the term K m + [S] is essentiallyequal to [S]. Replacing K m + [S] with [S] reduces equation(29) toV Svi= max[ ]Km+ [ S]V Sv ≈ max[ ]i ≈Vmax[S](31)Thus, when [S] greatly exceeds K m , the reaction velocityis maximal (V max ) and unaffected by further increases insubstrate concentration.(3) When [S] = K m (point B in Figures 8–3 and8–4).V S Svi= max[ ] V= max[ ] V= maxKm+ [ S]2[ S]2(32)Equation (32) states that when [S] equals K m , the initialvelocity is half-maximal. Equation (32) also reveals thatK m is—and may be determined experimentally from—the substrate concentration at which the initial velocityis half-maximal.A Linear Form of the Michaelis-MentenEquation Is Used to Determine K m & V maxThe direct measurement of the numeric value of V maxand therefore the calculation of K m often requires impracticallyhigh concentrations of substrate to achievesaturating conditions. A linear form of the Michaelis-Menten equation circumvents this difficulty and permitsV max and K m to be extrapolated from initial velocitydata obtained at less than saturating concentrationsof substrate. Starting with equation (29),Svi = Vmax [ ]Km+ [ S](29)