2 Enzymes

• Catalysts only affect reaction rates (not equilibria)
• Enzymes catalyse by stabilizing transition states and thus lowering activation barriers
• Steady-state studies involve conditions in which [S] >> [E], so that v can be measured over a reasonable period
• (EXPLAINED) Steady-state studies involve conditions in which the concentration of the substrate is much higher than the concentration of the enzyme, so that rate can be measured over a reasonable period
• The useful parameters we can derive from using the Michaelis-Menten equation: Affinity, Selectivity, Efficiency, Inhibitor characteristics - pharmacology
• Measuring the rates of enzyme-catalysed reactions:

  1. Keep pH and T constant
  2. Continuous assays preferred (e.g. spectrophotometry), not samples taken at multiple time-points.
  3. Conditions where [E] is low and [S] >> [E], allowing the enzyme to turn over many times
• Steady-state kinetics: The rate declines as substrate is depleted so measure early while v is constant and the system is in ‘steady-state’. The enzyme is also present but in limiting (low) concentrations.
• Steady-state rate (v) is proportional to [E] 
• With [E] constant, at a lower [S] the rate of reaction v is proportional to [S] but at higher [S] v becomes constant. The enzyme becomes saturated at high [S], giving Vmax
• At high [S], all enzyme is converted to the ES further increases in [S] do not increase rate. In these conditions, the reaction rate is limited by how quickly the ES complex (Michaelis complex) can produce and liberate the product and regenerate free enzyme= Vmax
• E + S    <->     ES    ->   E + P
• chymotrypsin + protein   <->  chymotrypsin:protein    -> chymotrypsin + peptides
• K is a dissociation constant. K = [E] [S] / [ES]
• ka is the forward rate constant, kd the backward rate constant
• The rate of appearance of ES = ka.[E].[S]
• The rate of disappearance of ES = kd.[ES]
• K = kd/ka
• In most cases the conversion of ES to E + P is the slowest step and is therefore rate-limiting. The rate constant for this reaction is kcat. The rate of appearance of product = kcat[ES]
• EQUATION 1: v = kcat[ES]
• at Vmax   [ES] = [ETotal]
• EQUATION 2: Vmax = kcat [ETotal] 
• EQUATION 3: [ES] = [E].[S]/K
• Relatively little of S is tied up in the complex ES, since [Stotal] >> [E], so [S] = [Stotal] which we know
• To find enzyme concentration: [E] = [Etotal] – [ES]
• EQUATION 4: [ES] = [Etotal].[Stotal]/(K + [Stotal])    
• EQUATION 5: v = kcat.[Etotal].[Stotal]/(K + [Stotal])    
• The Michaelis-Menten equation (EQUATION 6): v = Vmax.[S]/(K + [S])

  (we define the dissociation constant K as the Michaelis constant Km under these conditions)
• The Michaelis-Menten equation is a rectangular hyperbola.

  y = a.x/(b+x)

  The two constants (a) and (b) that define the hyperbola are Vmax and Km and x is [S]
• v =Vmax.[S]/(Km + [S]) is the Michaelis-Menten equation
• 1/2 Vmax is Km
• Km is given by the value of [S] at which v = Vmax/2
• Vmax  =  maximum velocity; Vmax = kcat [ET]
• Km  =  Michaelis constant
• Km is a measure of affinity
• ‘Enzyme efficiency’ is represented as :  kcat/Km .

  Its units are: s-1M-1
• Carbonic anhydrase (Substrate HCO3-) Km = 26mM
• Hexokinase (ATP) Km = 0.4 mM
• Hexokinase (Glucose) Km = 0.05 mM
• Vmax will vary with [E]
• If one enzyme accepts two different substrates, under identical conditions and the same [E], Vmax is the same, but Km may vary.

  Hence, the enzyme will have different affinities for different substrates