Page 285 - Introduction to chemical reaction engineering and kinetics

266 Chapter 10: Biochemical Reactions: Enzyme Kinetics

With V,,,, the maximum reaction velocity, replacing krcEo, equation 10.2-5 is rewrit-
ten as:

(10.2-9)
‘.’ = K,,, + cs

Equation 10.2-9 is known as the Michaelis-Menten equation. It represents the kinetics
of many simple enzyme-catalyzed reactions which involve a single substrate (or if other
substrates are in large excess). The interpretation of K,,, as an equilibrium constant is
not universally valid, since the assumption that step (1) is a fast equilibrium process
often does not hold. An extension of the treatment involving a modified interpretation
of K,,, is given next.

10.2.2 Briggs-Haldane Model
Briggs and Haldane (1925) proposed an alternative mathematical description of en-
zyme kinetics which has proved to be more general. The Briggs-Haldane model is based
upon the assumption that, after a short initial startup period, the concentration of the
enzyme-substrate complex is in a pseudo-steady state. Derivation of the model is based
upon material balances written for each of the four species S, E, ES, and P
For a constant-volume batch reactor operated at constant T and pH, an exact solu-
tion can be obtained numerically (but not analytically) from the two-step mechanism in
Section 10.2.1 for the concentrations of the four species S, E, ES, and P as functions of
time t , without the assumptions of “fast” and “slow” steps. An approximate analytical
solution, in the form of a rate law, can be obtained, applicable to this and other reactor
types, by use of the stationary-state hypothesis (SSH). We consider these in turn.
From material balances on S as “free” substrate, ES, E, and S as total substrate, we
obtain the following four independent equations, for any time t :

rs = dc,ldt = k-,c,, - klcscE (10.2-10)

rns = dc,,ldt = klcscE - kklCEs - k,cEs (10.2-11)

CEO = CE + CES (10.2-2)
CC&, = CS + CES + cp (10.2-12)

Elimination of cn from the first two equations using 10.2-2 results in two simultaneous
first-order differential equations to solve for cs and ens as functions of t:

dc,ldt = k-,C,, - klcS(cEo - cES) (10.2-13)

dc,,/dt = klCs(CEo - CES) - (k-, + k&s (10.2-14)
with the two initial conditions:

0 These can be solved numerically (e.g., with the E-Z Solve software), and the results
(10.2-15)
=
and ens
at t = 0,
0
= cso
53
v
used to obtain c&t) and cp(t) from equations 10.2-2 and 10.2-12, respectively.
For an approximate solution resulting in an analytical form of rate law, we use the
SSH applied to the intermediate complex ES. (If cnO << csO, the approximation is close