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9.17: Untitled Page 233

  • Page ID
    18366
  • Chapter 9

    the condition of local reaction equilibrium might be referred to as the steady‐state assumption for batch reactors.

    Section 9.2

    9‐8. In our study of Michaelis‐Menten kinetics we simplified the analysis by ignoring the influence of the substrate B, or any other substrate, on the enzyme catalyzed reaction of species A. When multiple substrates are considered, the analysis becomes very complex (Rodgers and Gibon, 1995); however, if we assume that the reversible binding steps are at equilibrium, the analysis of two substrates becomes quite tractable. The binding between enzyme E and substrate A is described by

    k I

    E A

    

    

    EA

    (1)

    k II

    and when A is in equilibrium with EA we can replace Eqs. 9‐64 and 9‐65 with the equilibrium relation given by

    k I

    I

    c

    c c

    K c c

    (2)

    EA

    E A

    eq E A

    k II

    The reversible binding between enzyme E and substrate B is similarly described by

    E B

    B

    E

    (3)

    and we express the equilibrium relation as

    II

    c

    K c c

    (4)

    EB

    eq E B

    In this model we assume that once substrate B has reacted with enzyme E to form the complex EB, no additional reaction with substrate A can take place. In this step substrate B acts as an inhibitor since it removes some enzyme E from the system. However, the complex EA can further react with substrate B to form the complex EAB as indicated by

    EA B EAB

    (5)

    The equilibrium relation associated with this process is given by III

    c

    K c c

    (6)

    EAB

    eq

    EA B

    Reaction Kinetics

    435

    In this step substrate B acts as a reactant since it produces the complex EAB that is the source of the product D.

    Because of the assumed equilibrium relations indicated by Eqs. 2, 4 and 6, we have only a single rate equation based on an irreversible reaction. This irreversible reaction is described by

    k

    Elementary chemical kinetic schema III:

    III

    EAB 

    E D

    (7a)

    Elementary stoichiometry III:

    III

    III

    III

    III

    R

      R , R

    R

    (7b)

    EAB

    E

    D

    E

    Elementary chemical kinetic rate equation III:

    III

    R

    k c

    (7c)

    D

    III EAB

    Since the product D is involved in only this single reaction, we express the rate of production as

    R

    k c

    (8)

    D

    III EAB

    In the absence of significant cell death, one can assume that the enzyme E

    remains within the cells, thus the total concentration of enzyme is constant as indicated by

    o

    c

    c

    c

    c

    c

    (9)

    E

    EA

    EB

    EAB

    E

    In this problem you are asked to show that Eqs. 2 through 9 lead to an equation having the form of

    c

    

    c

    A

    B

     

    

    D

    R

    max

    (10)

    K

    A

    cA

    K

     B

    cB

    provided that you impose the special condition given by

    III

    II

    K

    K

    (11)

    eq

    eq

    This restriction is based on the idea that B binds with EA in the same manner that B binds with E.

    9‐9. In Problem 9‐8 we considered a case in which the substrate B acted both as a reactant in the production of species D and as an inhibitor in that process. In some cases we can have an analog of the substrate that acts as a pure inhibitor and the Michaelis‐Menten process takes the form

    436