Page 288 - Introduction to chemical reaction engineering and kinetics

10.4 Inhibition and Activation in Enzyme Reactions 269

10.3.2 Linearized Form of the Integrated Michaelis-Menten Equation
For a constant-volume BR, integration of the Michaelis-Menten equation leads to a
form that can also be linearized. Thus, from equation 10.2-9,

rp = (-rs) = -% = VmaxCS (10.3-3)
Km + cs
or

K +c,
Ldcs = -V,,,,, dt
CS
or

hdcs + dcs = -V,,,,, dt (10.34
CS
which, with the boundary condition cs = csO at t = 0, integrates to

(100.3-6)
Km ln(cdcd + (cs - d = - Laxt
Equation 10.3-6 may be written as

t
1
Vmx
Wdcd =---- (10.3-7)
cso - cs Km Km cso - cs
According to equation 10.3-7, In (csIcsO)I(csO - cs) is a linear function of tl(csO - cs).
Km and Vmax can be determined from equation 10.3-7 with values of cs measured as a
function oft for a given csO.

10.3.3 Nonlinear Treatment

A major limitation of the linearized forms of the Michaelis-Menten equation is that
none provides accurate estimates of both Km and V,,,. Furthermore, it is impossible
to obtain meaningful error estimates for the parameters, since linear regression is not
strictly appropriate. With the advent of more sophisticated computer tools, there is an
increasing trend toward using the integrated rate equation and nonlinear regression
analysis to estimate Km and V,,,. While this type of analysis is more complex than the
linear approaches, it has several benefits. First, accurate nonbiased estimates of K,,, and
V,,, can be obtained. Second, nonlinear regression may allow the errors (or confidence
intervals) of the parameter estimates to be determined.
To determine K,,, and V,,,,,, experimental data for cs versus t are compared with val-
predicted by numerical integration of equation 10.3-3; estimates of K,,, and
0 Vmax are subsequently adjusted until the sum of the squared residuals is minimized.
ues of cs
V
The E-Z Solve software may be used for this purpose. This method also applies to other
“O-v
complex rate expressions, such as Langmuir-Hinshelwood rate laws (Chapter 8).
10.4 INHIBITION AND ACTIVATION IN ENZYME REACTIONS
The simple Michaelis-Menten model does not deal with all aspects of enzyme-catalyzed
reactions. The model must be modified to treat the phenomena of inhibition and