Part Three: Enzyme Kinetics to Understand the Mechanism

• Enzyme kinetics is the discipline focused on determining the rate of a reaction and how it changes in response to changes in experimental parameters.
• The pre steady state is the initial transient period during which ES builds up.
• Steady state is the period during which [ES] and other intermediates remain constant.
• Steady state kinetics in the traditional analysis of reaction rates.
• The initial rate or velocity, V_0, is the tangent to each curve taken at time zero. It reflects the steady state.
• At the beginning of the reaction, [S] is regarded as constant.
• The plateau like V_0 region is close to the maximum velocity, V_max.
• In the general theory of enzyme action proposed by Michaelis and Menten, in step one, enzyme and substrate combine to form a complex in a reversible, relatively fast step. In step two, the complex breaks down to yield the free enzyme and the reaction product in a slower step. Because this second step limits the overall reaction rate, the overall rate is proportional to [ES].
• The equation for the first step of Michaelis and Menten's enzyme action theory is E + S equilibrium ES where the forward reaction is k_1 and the reverse reaction is k_-1.
• The equation for the second step of Michaelis and Menton's enzyme action theory is ES equilibrium E + P where the forward reaction is k_2 and the reverse reaction is k_-2.
• V_max is observed when virtually all the enzyme is present as the ES complex. Here, further increases in [S] have no effect on the rate.
• V_max is responsible for the plateau observed.
• The curve expressing the relationship between [S] and V_0 can be expressed by the Michealis-Menten equation V_0 = (V_max X [S]) / (K_m + [S]) where V_0 is the initial velocity, V_max is the maximum velocity, [S] is the initial substrate concentration, and K_m is the Michaelis constant.
• The Michaelis Menten equation is the rate equation for a one substrate enzyme catalyzed reaction where V_0, V_max, [S] and K_m are readily measured experimentally.
• The overall reaction for the formation and breakdown of ES simplifies to E + S --> (k_1) <-- (k_-1) ES --> (k_2) E + P.
• V_0 is determined by the breakdown of ES to form product, which is determined by [ES]. V_0 = k_2 [ES].
• [ES] is not easily measured experimentally.
• [E_t] is the total enzyme concentration or the sum of free and substrate bound enzyme.
• Free or unbound enzyme is [E] = [E_t] - [ES]
• Because [S] is greater than [E_t], the amount of substrate bound by the enzyme at any given time is negligible compared with the total [S].
• Deriving the Michaelis Menten Equation

  • k_1 ([E_t] - [ES]) X [S]
  • k_-1 [ES] + k_2 [ES]
  • k_1 ([E_t] - [ES]) X [S] = k_-1 [ES] + k_2 [ES]
  • k_1 [E_t] [S] - k_1 [ES][S] = (k_-1 + k_2) [ES]
  • k_1 [E_t] [S] = k_-1 [ES] [S] + (k_-1 + k_2) [ES]
  • k_1 [E_t] [S] = (k_1 [S] + k_-1 + k_2) [ES]
  • [ES] = (k_1 [E_t] [S]) / (k_1 [S] + k_-1 + k_2)
  • [ES] = ([E_t] [S]) / ( [S] + {k_-1 + k_2} / k_1)
  • [ES] = ( [E_t] [S] ) / ( K_m + [S])
  • V_0 = (k_2 [ES] [S]) / (K_m + [S])
  • V_0 = (V_max [S]) / (K_m + [S])
• Rate of ES formation is k_1 ( [E_t] - [ES]) [S]
• The rate of ES breakdown is k_-1 [ES] + k_2 [ES].
• The steady state assumption is that the rate of formation of ES is equal to the rate of its breakdown.
• Michaelis constant, K_m , is ( k_-1 + k_2) / k_1
• V_0 = k_2 [ES]
• Given that V_max occurs when the enzyme is saturated (when [ES] = [E_t]), V_max = k_2 [E_t]
• When V_0 = 1/2 V_max

  • V_max / 2 = (V_max [S]) / (K_m + [S])
  • 1/2 = [S] / (K_m + [S])
  • K_m = [S] when V_0 = (1/2) V_max
• An algebraic transformation of the Michaelis Menten equation converts the hyperbolic curve into linear form.

  • V_0 = (V_max [S]) / (K_m + [S])
  • (1 / V_0) = (K_m + [S]) / (V_max [S]).
• Deriving the Lineweaver Buck equation

  • (1/ V_0) = (K_m + [S]) / (V_max [S])
  • (1 / V_0) = (K_m) / (V_max [S]) + [S] / (V_max [S])
  • (1 / V_0) = (K_m) / (V_max [S]) + 1 / V_max
• For enzymes obeying the Michealis Menten relationship, a Lineweaver Burk (Double reciprocal) plot, or a plot of 1 / V_0 versus 1 / [S], yields a straight line.
• All enzymes that exhibit a hyperbolic dependence of V_0 on [S] follow a Michaelis Meneten kinetics where

  K_m = [S] when V_0 = (1/2) V_max
• K_m can vary for different substrates of the same enzyme.
• For reactions with two steps, K_m = (k_2 + k_-1) / k_1.
• For the equation, K_m = (k_2 + k_-1) / k_1, when k_2 is rate limiting, k_2 << k_-1, and K_m reduces to k_-1 / k_1, which is defined as the dissociation constant, K_d of the ES complex. Under these conditions, K_m represents a measure of affinity.
• The number of reactions steps and the identity of the rate limiting steps varies from enzyme to enzyme.
• For a two step Michaelis Menten mechanism, V_max = K_2 [E_t].
• When there is product release, that is the rate limiting step. This equation would EP --> E + P.
• The general rate constant, k_cat, describes the limiting rate of any enzyme catalyzed reaction at saturation.
• If one step in a multistep reaction is clearly limiting, k_cat equals the rate constant for the step. But, it is more complex when several steps are rate limiting.