# 9.8: Untitled Page 224

- Page ID
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## Chapter 9

and use of the constraint on the total concentration of enzyme given by Eq. 9‐70

provides

o

*k *( *c * *c*

) *c*

I

*E*

*EA*

*A*

*cEA*

*k *

(9‐74)

*k*

II

III

Solving for the concentration of the enzyme complex gives

o

*c c*

*E*

*A*

*c*

(9‐75)

*EA*

( *k * *k *) *k * *c*

II

III

I

*A*

This result can be used in Eq. 9‐68 to represent the net rate of production of the desired product as

o

( *k*

*c *) *c*

III

*E*

*A*

*R*

*D*

*K*

(9‐76)

*c*

*A*

*A*

in which *K * is defined by

*A*

*K*

( *k * *k *) *k *

(9‐77)

*A*

II

III

I

The maximum net rate of production of species *D* occurs when *c * *K * and *A*

*A*

this suggests that Eq. 9‐76 can be expressed as

*max cA*

Michaelis‐Menten kinetics:

*D*

*R*

(9‐78)

*K*

*c*

*A*

*A*

This *microscopic* result is identical *in form* to the *macroscopic* Monod equation for cell mass production (see Eq. 8‐58); however, the production of cells shown earlier in Figure 8‐8 is not the same as the production of species *D* illustrated in Figure 9‐9. Certainly there is a connection between the production of cells and the production of intercellular material, and this connection has been explored by Ramkrishna and Song (2008).

It is of some interest to note that when the rate of production of species *D* is completely controlled by the reaction illustrated by Eq. 9‐66a, we have a situation in which

*k*

*k *

(9‐79)

III

II

and the parameter *K * in Eq. 9‐77 simplifies to

*A*

1

*K*

*k*

*k*

*K*

*, *

*k*

*k *

(9‐80)

*A*

II

I

*eq*

III

II

417

In this case *K * becomes the inverse of a true *equilibrium coefficient*. Since the *A*

imposition of Eq. 9‐79 has no effect on the *form* of Eq. 9‐78 there is often confusion concerning the precise nature of *K *.

*A*

**9.3 Mechanistic matrix **

In this section we explore in more detail the reaction rates associated with chemical kinetic schemata of the type studied in the previous two sections. The *mechanistic matrix* (Björnbom, 1977) will be introduced as a convenient method of organizing information about reaction rates. This matrix is different than the *pivot matrix* discussed in Chapter 6, and we need to be very clear about the similarities and differences between these two matrices, both of which contain coefficients that are often referred to as *stoichiometric coefficients*. In some cases the mechanistic matrix is identical to the *stoichiometric matrix* and in some cases it consists of both a *stoichiometric matrix* and a *Bodenstein matrix*.

We begin by considering a system in which there are five species and three chemical kinetic schemata described by

Elementary chemical kinetic schema I:

*k* I

*A * *B * *C * *D , *

*D * is a by‐product

(9‐81a)

Elementary chemical kinetic schema II:

*k* II

*C * *B *

*E , *

*E * is the product

(9‐81b)

Elementary chemical kinetic schema III:

*k* III

*C * *D *

*A * *B , * reverse of schema I

(9‐81c)

In this example we *assume* that the *stoichiometric schemata* are identical in form to the *chemical kinetic schemata*, and we carefully follow the structure outlined in Sec. 9.1.1 in order to avoid algebraic errors. We begin with the first elementary step indicated by Eq. 9‐81a, and our analysis of this step leads to *k* I

Elementary chemical kinetic schema I:

*A * *B * *C * *D *

(9‐82a)

I

I

I

I

I

Elementary stoichiometry I:

I

*R*

*R , *

*R*

*R , *

*R*

*R * (9‐82b)

*A*

*B*

*A*

*C*

*A*

*D*

418