Kinetics and Regulation of Enzyme Catalysis
What is enzyme catalysis?  Acceleration of the rate of a chemical reaction by the stabilization of the transition-state complex.  Stability is measured in terms of the free energy, which is derived from the chemical bonding energy (heat of formation) and the entropy.  Enzymes cannot alter the heat of formation, therefore, they must alter the entropy.  There are two practical effects of enzymes:  1)  raising the effective concentration of reactants, and 2) decreasing the entropy of the reactants.  Decrease in entropy is provided by structural constraints and solvent shielding.  The net effect of stabilizing the transition state is that chemical reactions that would take a very long time to occur without assistance are very fast.  How fast?
How do you measure the rate of an enzyme-catalyzed reaction?  Enzyme catalysis is detected by measuring either the appearance of product or disappearance of reactants.  To measure something, you must be able to see it.  Enzyme assays are tests developed to measure enzyme activity by measuring the change in concentration of a detectable substance.  The direct assay would measure the appearance of a product that absorbs or emits detectable energy (UV/vis absorbance, fluorescence, phosphorescence, radiation); or would measure the disappearance of a substrate that emits or absorbs detectable energy.  In an indirect assay, the conversion of S to P by the enzyme of interest could be coup
led to a second reaction (chemical or enzyme-catalyzed) that uses the product of the first reaction as a substrate for the formation of a detectable product.
Definition:  Assay (from the French, to try) a test that measures the concentration of a specific molecule of interest.
Enzyme assays may be used to determine the concentration of a limited amount of reaction substrate by converting all of it into a detectable product, in which case one substrate is limiting, and enzyme is in excess.  Enzyme assays that are used to determine the concentration of the enzyme have all substrates in excess and a small, unknown, amount of enzyme.  Enzyme assays may also be used to study the mechanism of catalysis by limiting the concentration of both enzyme and substrate.  Enzymes are frequently hard to isolate, so in mixtures of proteins, a single enzyme is detected by measuring its activity.  An international unit of activity is one umole product formed per minute U = µmol P/min
k1k3
E + S                              ES                            E + P
k2k4
Michaelis-Menten kinetics are defined by a hyperbolic saturation curve when initial velocity, V0 , is plotted against substrate concentration, [S].  When the rate at which substrate is consumed is directly proportional to the remaining concentration of substrate, the reaction follows first-order kinetics.  In first-order kinetics, a constant proportion of present substrate is consumed per unit time, NOT a constant amount of substrate.  This is described mathematically as shown:
–d S
=  V0= K M  [S]
------
d t
Amount of  in other words, is some of the substrate
that
at S consumed initial velocity of  constant  present
per unit time the reaction fraction instant in time
OR
d P
=  V0= K M  [S]
------
d t
Amount of  in other words, is some of the substrate
that
at P formed initial velocity of  constant  present
per unit time the reaction fraction instant in time
The first-order process is logarithmic because the reaction velocity is constantly changing
with changing [S].  The plot of initial reaction velocity vs. [S] is hyperbolic because the velocity increase
s proportionately with substrate concentration (first-order); however, with further increases in substrate concentration, the rate of increase in the velocity slows down until it reaches a maximum value limited by the concentration of enzyme.  At this point it displays zero-order kinetics; the rate is constant.  Michaelis -Menten kinetics are confined to the initial, first-order part of the curve.  Michaelis-Menten kinetics are displayed when [ES] is constant and at equilibrium, and there are not multiple forms of the ES transition state.  In complex systems, [ES] may be constant (steady-state) but not at its equilibrium concentration as defined by rate constants of the reaction equation.
initial conditions valid  Zero-order;  rate independent of [S]
= k3)
Great, but we can’t measure [E] or [ES] easily.  We can measure V0 as ∂P/∂t.  We defined the first-order reaction velocity, V0 = K M [S], but we are trying to put a value on K M.
Express initial rate of product formation V0 using the rate equation:
V0  = k3 [ES]
V max  = k3 [E]T when the enzyme is fully occupied, saturated
when not saturated, [E]T = [E] + [ES]
Substitute into the definition of K M terms we can measure for those we cannot. rearrange expressions of [E]T , V0 and V max: [E] = [E]T – [ES]  Eqn1
2
[E]T = V max / k3Eqn
3
[ES] = V0 / k3Eqn Start with the definition of K M
K M    =    [E] [S]
[ES]
use Eqn 1 above to get:
K M=([E]T – [ES]) ([S])
[ES]
multiply to get:
reaction orderK M=  [E]T [S]  –  [ES] [S]
[ES]      [ES]
cancel [ES] and use Eqn 2 to get:
K M=V max [S]  –  [S]
k3 [ES]
use Eqn 3 to get:
K M=V max k3 [S]  –  [S]
k3 V0
cancel k3
K M=V max [S]  –  [S]
V0
We can measure V max, V0, and [S]!  Now rearrange:
K M  + [S] = V max [S]
V0=V max [S] So, the initial velocity is max velocity times factor based on [S] K M  + [S]
M

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