Cell Metabolism - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 494

k 3
k 3

E O $ E R þ 4 S L % E O $ E R $ð S L Þ 4 (10.7)

k 4

E O þ E R $ð S L Þ 4 % E O $ E R $ð S L Þ 4 (10.8)

k 4

The model in the above set of eqns (10.5) through (10.8) gives a simplified description of the
true system since there may be different binding affinities for the repressor protein depend-
ing on how much lactose is bound to the protein. With the concentration of the species (indi-
cated with squared brackets) being in moles per gram dry weight, the (reciprocals of
equilibrium) constants K i , i

¼ 1, 2, 3, 4 are given by:

S L 4

K 1 ¼ k 1
k 1

½

E R ½

¼ (10.9)

½ E R $ð S L Þ 4

K 2 ¼ k 2
k 2

¼ ½

E R ½

O

E

(10.10)

½ E

O $ E R

S L 4

K 3 ¼ k 3
k 3

½ E

O $ E R ½

¼ (10.11)

½ E

O $ E R $ð

S L Þ 4

K 4 ¼ k 4
k 4

¼ ½

E

O ½

E R $ð S L Þ 4

(10.12)

½ E

O $ E R $ð

S L Þ 4

A macroscopic description can be used to express the influence of the reacting species on
the kinetics, i.e. the concentrations of the different components are used. However, microor-
ganisms only contain a few (1 e 4) copies of one type of operator per cell and the number of
repressor proteins per cell is also low (10 e 20). For such small entities, the meaning of concen-
trations and of thermodynamic equilibrium is disputable, and it may be more correct to
apply a stochastic modeling approach. As in Michaelis e Menten kinetics for enzymes all reac-
tions in (10.6) through (10.8) are assumed to be equilibrium reactions. This is reasonable since
the relaxation times for the equilibria are much smaller than for most other cellular reactions.
Balances for the repressor, operator, and inducer are

½ E R T ¼½ E R þ½ E R $ð S L Þ 4 þ½ E O $ E R þ½ E O $ E R $ð S L Þ 4 (10.13)

½

E O T ¼½ E O þ½ E O $ E R þ½ E O $ E R $ð S L Þ 4

(10.14)

½ S L T ¼½ S L þ4 ½ E R $ð S L Þ 4 þ4 ½ E O $ E R $ð S L Þ 4 (10.15)

where the subscript T refers to the total concentration. In wild-type E. coli, there are 10 e 20
times more repressor molecules than there are operators, and in this case, the last two terms
in Eqn (10.13) can be neglected. That is,

½ E R T z ½ E R þ½ E R $ð S L Þ 4 ¼½ E R þK 1 ½ E R ½ S L (10.16)

4

Equation (10.15) can be simplified by assuming that the intracellular concentration of
inducer molecules is in sufficient excess over repressor molecules, and consequently that
4[E R $ (S L ) 4 ]

þ

4[E O $ E R $ (S L ) 4 ]

<<

[S L ]. That is,
½

S L z ½ S L T

(10.17)