Module 6 : Reaction Kinetics and Dynamics
Lecture 28 : Elementary Reactions and Reaction Mechanisms
  28.5

Enzymes Kinetics:

Enzymes are biological catalysts which help in converting substartes (S) into products (P). We will now describe the Michaelis - Menten mechanism for enzyme action. Initially, the enzyme ( E ) and the substrate form a complex E S* which later decays into the products regenerating the enzyme. The reaction scheme is written as

 

E + S ES * P + S

(28.39)
The rate of formation of the product is


d [ P ] / d t = k 2 [ E S * ] (28.40)
Using the steady state approximation for [ E S *], we have
 
d [ E S * ] / d t = k f [ E ] [ S] - k r [ E S * ] - k f [ E S * ] = 0 (28.41)
The first term gives the rate of formation of E s * and the other two terms give the rate of decay of E S *
 
[ E S * ] = k f [ E ] [ S ] / ( k r + k 2) (28.42)
If [ E ]0 is the initial (total) concentration of the enzyme and [ E ] is the concentration of the enzyme which is free to react, then [ E ]0 = [ E ] + [ E S *] which equals the free plus the bound enzyme which has to be equal to [ E ]0 usually [ E ]0 is much less than [ S ] , and so [ S ] [ S ] 0 = [ S ] total. Substituting in we get
 
[ E S * ] = k f / k r + k 2 [ S ] { [ E ] 0 - [ E S * ] } (28.43)
 
Rearranging, [ E S * ] = k f [ E 0] [ S ] / k 2 + k r + k f [ S ] (28.44)
 
and d [ P] / dt = k 2 k f [ E 0 ] [ S ] / k 2 + k r + k f [ S ] = k 2 [ E ] 0 [ S ] / kM + [ S ] (28.45)
Where kM = ( k 2 + k r ) / k f is the Michaelis constant. We see that the rate of enzymolysis depends linearly on [ E ]0 but in a more involved way on [ S ]. When [ S ] is much larger than kM, we have d [ P ] /d t = k 2 [ E ] 0, independent of the substarte and is thus a zero order reaction (in S) . This is because too much of S is present to be affected by its depletion into products!
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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