APPENDIX I Fundamentals of Stoichiometry of Complex Reaction Systems Bioprocessing includes a large variety of metabolic reactions even on the macroscopic level (cf. Fig. 2.16) and at the same time is carried out in different modes of reactor operations. Therefore, complex reaction systems must be treated stoichiometrically, which means that not only complex reactions themselves must be considered but also complex reactor operations. 1.1 Stoichiometry of Complex Reactions The stoichiometric equation in the case of complex reactions can be written in the form while for a simple reaction the following form is valid: where i = number of reactions (1 :::;; i :::;; M) j = number of components (1 :::;; j :::;; N) Aj = components of reaction mixture v = stoichiometric coefficients (U) (1.2) The differential change of the number of moles n of component Aj due to the reaction i is defined as (1.3) where ei is the extent of reaction, defined as the change of number of moles divided by the stoichiometric coefficient [mole]. In complex reactions, the singular reaction steps are interconnected, that is, singular components participate in different reactions, with the conse- quence that a part of the reactions is stoichiometrically dependent. Only the independent reactions can be determined from the change in the number of moles. The solution to the problem of stoichiometric dependence can be found
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APPENDIX I
Fundamentals of Stoichiometry of Complex Reaction Systems
Bioprocessing includes a large variety of metabolic reactions even on the macroscopic level (cf. Fig. 2.16) and at the same time is carried out in different modes of reactor operations. Therefore, complex reaction systems must be treated stoichiometrically, which means that not only complex reactions themselves must be considered but also complex reactor operations.
1.1 Stoichiometry of Complex Reactions
The stoichiometric equation in the case of complex reactions can be written in the form
while for a simple reaction the following form is valid:
where i = number of reactions (1 :::;; i :::;; M) j = number of components (1 :::;; j :::;; N)
Aj = components of reaction mixture v = stoichiometric coefficients
(U)
(1.2)
The differential change of the number of moles n of component Aj due to the reaction i is defined as
(1.3)
where ei is the extent of reaction, defined as the change of number of moles divided by the stoichiometric coefficient [mole].
In complex reactions, the singular reaction steps are interconnected, that is, singular components participate in different reactions, with the consequence that a part of the reactions is stoichiometrically dependent. Only the independent reactions can be determined from the change in the number of moles. The solution to the problem of stoichiometric dependence can be found
1.1 Stoichiometry of Complex Reactions 407
with the aid ofthe "matrix of the stoichiometrical coefficients" of the following structure
~ Components 1 ~ j ~ N
~ VI
VI (1.4)
where the row index is the number of reactions i and the column index is the number of components j. Each row of the stoichiometric coefficient matrix expresses the stoichiometry of a reaction in terms of the number of moles of each compound converted per unit reaction rate.
The number of stoichiometrically independent reactions is given by the rank of the matrix R p, which can be determined with e.g. the aid of the Gaussian method of elimination. As a result, the stoichiometrical coefficients of Rv linearly independent equations for the reaction system are necessary and sufficient for, for example, calculation of the conversion of the key variables and therefore also for all other components. Thus
(1.5)
Sometimes balancing is carried out without the formulation of reaction equations. This situation, however, arises rarely when kinetics are of interest. In this case an "element -species matrix" can be written on the basis of N species (components) with k elementary balances
k'S1 Components I ~ j ~ N
..::.: all .... · ...... VI
VI a21 ... · .... ,. ..
en ....... ::: Cl)
E ~l ...... · ......
Cl)
G:l
where row index = number of components, j column index = number of chemical elements, i
alj
a2j
aij
i = number of elementary balances (1 ::;; i ::;; k)
(1.6)
408 I. Fundamentals of Stoichiometry of Complex Reaction Systems
aij = number of atoms of atomic species i present in molecule of componentj «resp. aj' bj , cj ' dj , etc.)
The number of key variables R is determined again with the aid of the rank of this matrix R. Thus (Schubert and Hofmann, 1975)
R = N - Rp (1.7)
Normally Rp = k in the case of N species and k elements. Hence, only (N - k) net conversion rates can be chosen independently.
From this reasoning it becomes clear that the number of independent kinetic equations to be postulated cannot be chosen at will-it is completely specified by the number of elementary balances k and the number of components N in the system (Roels, 1980). Which key variable or which kinetic equation is to be chosen strongly depends on the application one has in mind.
As an example for the determination of the number of independent equations, the situation of an organism with balanced growth is considered. This organism grows on one sole source of carbon and energy, a source that may contain nitrogen. One sole source of nitrogen is supplied, and this source may also contain carbon. One product is excreted; CO2 , H 2 0, and O2 are the only other components, to be exchanged with the environment.
In terms of formalism, the system of organisms will be considered to be a given quantity of mass. The organism exchanges, macroscopically speaking, an exact replica of itself with the environment; it is characterized by its gross elemental composition formula Cal Hbl 0Cl Ndl . The concept of the C-mole of organism, that is, the amount containing 1 mole of carbon, is adopted (see Sect. 2.2.3.2). Figure 6.53 gives a schematic representation of the system and the possible flows to and from the system. Only the elements C, H, 0, and N are considered, and indeed these elements comprise in most cases about 95% of the cellular mass and the various other exchange flows. Equation 2.11 can be directly applied to this specific case, with Vn; in the case of stirred tank reactors being the rate of flow of components (Fj' C-mole/m3 hr).
The element-species matrix that represents the elemental composition of the flows of compounds can be written in this case as
k~1 C H ° N
F I : X bi ci d l
F 2 : Sc a2 b2 c2 d2
F3 : P a3 b3 c3 d3 (1.8) F 4 : SN a4 b4 c4 d4
F5: °2 0 0 2 0
F6: CO2 0 2 0
F7: H2O 0 2 0
1.2 Example for Determination of Number of Linearly Independent Equations 409
In the present case there are seven flows, and Equ. 2.11 specifies four equations between the flows represented in the matrix of Equ. 1.8. Hence, only three flows are independent variables (cf. Sect. 1.2). Which kind offlows to be chosen for measurement depends on the possibilities for experimental determination. The knowledge, for example, of the respiratory quotient and the ratio of oxygen consumption to substrate consumption allows direct estimation of the biomass production rate and the product formation rate. This conclusion from the application of balancing is of the greatest importance in situations where process variables, for example, X, are very difficult to measure, which is the case in penicillin fermentation (Mou and Cooney, 1983).
1.2 Example for Determination of Number of Linearly Independent Equations ("Key Reactions")
Reaction scheme
Reaction steps
VA 'CA + VBl'CB ~ VCl 'Cc
VCl 'Cc~ VA 'CA + VBl 'CB
VC2 • Cc + VB2 • CB ~ Vo' Co
Vo • CD ~ VC2 • Cc + VB2 • CB
(1.9a)
(1.9b)
(1.9c)
(I.9d)
(1.ge)
(1.9f)
Process kinetics includes four rates of individual steps (r 1 to r 4) and the rate equations for all compounds can be written on the basis ofrl to r4 as follows:
rC = V<;:l • r l - VCl • r2 - VC2 ' r3 + VC2' r4
ro = vO'r3 - vO'r4
Thus, the matrix of stoichiometric coefficients is:
o
(1.1Oa)
(1.10b)
(1.10c)
(1.10d)
(1.11)
The rank of this matrix can be shown to be two by using the method of Gauss
410 I. Fundamentals of Stoichiometry of Complex Reaction Systems
elimination or by finding the order ofthe determinant, which is not zero. Thus, for example, the second column is identical with the first when multiplied by -1; the same with the fourth and the third columns. The remaining matrix is then
(~ o 0) -Va2 0 -VC2 0
VD 0
(1.12)
With Rp = 2, the number of key reaction or key variables is two, and these can be chosen arbitrarily (e.g., A and D). Thus the remaining dependent rates can be written on the base of rA and rD :
(1.13a)
(1.13b)
1.3 Stoichiometry of Complex Reactor Operation
The majority of stoichiometric considerations is restricted to closed reactor operations, where div (cjv) = 0, so that according to Equ. 2.3b
dc· _J = +v .. r dt - J
(1.14)
A simple case appears also with continuous stirred tanks.
13.1 SEMIDISCONTINUOUS REACTOR
For semidiscontinuous processing, often used in fermentations for the production of yeast biomass or secondary metabolites, the basic balance equation is to be modified. Balancing in this case has to distinguish between components that are already present in the reactor at the beginning (number of moles nj,o) and components that are fed later on to the reactor (flux of moles liY>- The stoichiometric balance in this case of semi discontinuous reactor operation is
where nj = number of moles of component j njO = temporal initial value of n nY = spatial initial value of n
(1.15)
This operation contains the time t as an independent process variable, in contrast to all batch reactor configurations.
1.3 Stoichiometry of Complex l{eactor Operation 411
1.3.2 NONSTATIONARY REACTOR OPERATION
Stoichiometry, for example, in the case of nonstationary modes of reactor operation, needs another fundamental equation. For the CSTR the balance of a component j is written as
dc· c· - cf! d~. d/ + Y = f vij de'
which gives after integration
cj = cY + (cj,o - cY)e-t/ f + I vij' ~*(t) i
where ~* is the modified extent of reaction.
(1.16)
(1.17)
Characteristically this equation contains again the time t as a process variable. This term with the time t disappears and can be neglected in the case of stationarity (if t » T) and when cj,o = cy, that is, in a batch reactor. Therefore the stoichiometric balance equation of a discontinuous system can be formally applied to a nonstationary continuous stirred tank. Similarly, the stoichiometric equations of heterogeneous reactor systems with interfacial mass transfer can be derived (Budde, Bulle, and Riickauf, 1981).
BIBLIOGRAPHY
Budde, K., Bulle, H., and Riickauf, H. (1981). Stochiometrie chemisch-technologischer Prozesse. Berlin: Akademie Verlag.
Mou, D.G., and Cooney, Ch.L. (1983). Biotechnol. Bioeng., 25, 225, and 257. Roels, J.A. (1980). Biotechnol. Bioeng., 22, 2457. Schubert, E., and Hofmann, H. (1975). Chem. lng. Techn.,47, 191.
APPENDIX II
Computer Simulations*
This appendix contains a series of computer simulations that are thought to represent the most significant basic kinetic models in bioprocessing. The models are summarized in Table 11.1. The simulations in the figures also contain the values of model parameters chosen for demonstration. Mainly two different kinds of plots are presented, the first showing concentration/time curves and the second the corresponding time curves of specific rates of bioprocesses. The models are as follows:
1. Simple Monod-kinetics in batch operation:
Reaction scheme: S ~ X
Reactor balance equations (DCSTR):
rx = fl(S)' x
1 - rs = -' fl(S) . x
YX!S
with kinetic equation (Monod type, cf. Equ. 5.38):
S
fl(S) = flmax Ks + S
Model parameters; process variables: flmaX' Ks, YX!S; Xo, So· 2. Monod kinetics with lag time tL (cf. Sect. 5.3.3.1):
rx = fl(S, t)· x
1 -rs = -' fl(S,t)X
YX!S
(II.1 )
(1I.2)
(11.3)
(11.4)
(11.5)
(11.3)
*This appendix has been added to the English edition of this book as a consequence of a critical recommendation by 1.1. Dunn (Federal Technical University/ Lab of Technical Chern. Zurich). Most of the simulations were realized at Graz the University of Technology, Austria, or at ZIMET, Central Institute of Microbiology and Experimental Therapy, lena, East Germany (Fig. II.33 and II.34).
TA
BL
E I
LL
Ove
rvie
w o
f sim
ulat
ed b
iopr
oces
s ki
neti
c m
odel
s.
No.
B
iore
acto
r op
erat
ion
Pro
cess
var
iabl
es
Kin
etic
mod
el t
ype
Var
iatio
n E
quat
ions
F
igur
es
DC
ST
R
X,S
M
icro
bial
gro
wth
: S
o IL
l-II
.4
II.!
M
onad
X
O
II.2
Y X
IS
11.3
/lm
ax
II.4
K
s II
.S
2 D
CS
TR
X
,S
Mon
ad w
ith
tL
tL
II.S
, 1I
.6
II.6
3
DC
ST
R
X,S
M
onad
with
kd
kd
II.3
, II
.7
II.7
, II
.8
4 D
CS
TR
X
,S
Mo
nad
wit
h m
s m
s II
.2, 1
I.8
1I.9
, IU
O
5 D
CS
TR
X
,S
Mon
ad w
ith k
d,
ms
k d,
ms
II.7
, II
.S
11.1
1, 1
1.12
6
DC
ST
R
X,S
M
on
ad w
ith S
inhi
biti
on
K,s
II
.2, 1
I.3,
H.9
II
'!3,
II.
!4
7 D
CS
TR
X
,S
Mon
od w
ith
2-S
lim
itat
ion
KI2
II
.10-
II.1
3 11
.15
8 D
CS
TR
X
,S
Mon
ad:
diau
xie
Scr
it
11.1
4-II
.19
ILl6
, IL
l 7
KI2
(KR
) II
.1S,
II.
! 9
,....
kLia
H
.20
-II.
23
II.2
0, I
I.21
~
()
K,p
, K
,sp(
KR
),
II.2
4-I
1.29
II
.22-
II.2
4, n
.25
-II.
29
0
Kp,
K,s
x 3 '"d
9 D
CS
TR
X
,S,O
M
onad
wit
h O
2 li
mit
atio
n 10
D
CS
TR
X
,S,P
; m
ulti
inhi
biti
on m
odel
for
mic
robi
al
prod
ucti
on (
with
rep
ress
ion
and
I:: .....
mul
tiin
hibi
tion
s)
Fs
1I.2
4-11
.30
II.3
0, 1
1.31
<1
l ... 11
S
CS
TR
X
,S,P
T
ype
10 w
ith
S fe
ed
cJt
11.3
1-11
.38
II.3
2 t/
) a' S
o 11
.39,
11.
40 a
nd
11.3
3 I::
Xo
1I.4
II
.34
§: o·
12
DC
ST
R
Sc,
Sd'
Sad
s B
ioso
rpti
on
13
CS
TR
X
,S
Mon
od
::l
;;;l
en
~
-!:>-
,....
w
r-, <:> ~
S·
::: ~ '" ::=
414 II. Computer Simulations
x ... X (C) •• • S + . . MY C) . . . p MODEL : '1'10 00 '
SO 0 0 0
1 1 . S GI L ~ a: . 2 ) 1 . 0 GIL "' 3) 2.0 GIL
0 0 0 a: 0 ...
0 0 0 3 0 0 0
~ 3 0
0 0 0 ... :::: "' :::: ., M
Cl Cl I ....
Ul X " lil ill d
>- N L
2
ill " 0
'" .
'" 0' . ~
:5 ~ 0
"! . 00 1.00 q . DO 5.00 8.00 , .00 9.00 9 . 00 10 DO
~1 o ~ -=-=-=-=-=-=--'::-=--="-::::~'""2_=:::_==:=-.::..:-::.;--=-~-~-..:.- .::..- .::..- -:::.:-:..:-:.:-..:.-l:::===::;j
o FIG. II.33
X
(go(')
.5
o
.5 D
UL ______________ ~ ________________ u
o 1- l00h
FIG. II.34
II. Computer Simulations 431
with
Model parameters: Jlmax, Ks, Yx1s, tL. 3. Monod kinetics with death rate kd (cf. Sect. 5.3.4.1):
rx = Jl(s)x - kd . x
1 -rs = -' Jl(s)x
Yxis
with Equ. 11.4 for Jl(s) and Jlmax, Ks, Yxls, kd •
(II.6)
(11.7)
(II.3)
4. Monod kinetics with endogeneous metabolism (ms) (cf. Sect. 5.3.4.2):
rX = Jl(s)x
1 -rs = -Jl(s)x - ms' x
Yxis
with Equ. 11.4 for Jl(s) and Jlmax, Ks, Yxls, ms; xo, So·
(11.2)
(II.8)
5. Monod kinetics with kd and m. as a combination of Equs. II.7 and II.8. 6. Monod kinetics with S inhibition (cf. Sect. 5.3.5.1). Thus, taking Equs. II.2
and II.3 with Equ. 5.88:
1 Jl(s) = Jlmax 1 + Ks/s + s/K,s
and K,s being the key parameter. 7. Monod kinetics with double substrate limitation (cf. Sect. 5.5):
rx = Jl(S1' t)x + Jl(S2)X' f 1
-rS1 = --' Jl(S1, t)x YX1S1
1 - rS2 = --' Jl(S2)X . f
YX1S2
(11.9)
(11.10)
(II. 11)
(II. 12)
with Equ. II.6 for Jl(S1' t) and the following expression for the term f (cf. Equ. 5.106 or Equ. 5.152):
(II. 13)
with KR being the key parameter for repression and s1,crit representing the substrate concentration, where both substrates are utilized simultaneously, thus losing the diauxic behavior.
8. Diauxic growth as an analogy to case 7:
432 II. Computer Simulations
scheme: s.------"X 1------. P = S2 --+ X
rX = fl(Sl,t)X + fl(p)x'f1'f2 (II.14)
1 1 -rS1 = --' fl(Sl' t)x + -' qp(s)' X
YX1S1 YPls (11.15)
(11.16)
with kinetic equations
flS2 = J1m.x, 2 K + S2 S2
S2 (S2 == p) (II.I8)
(II.I9)
and Equ. II.I3 for the term fl' f2 = f This simplest model of diauxie thus contains 13 parameters: flm.x,l, J1max,2, K S1 ' Kp(== K S2 )' YXIS1 ' YxIP ' YPIS1 '
YXIS2 ' t Ll , K ls , KIP, KR (== K12 ), and sl,crit·
9. Monod with O2 limitation (cf. Sect. 5.5.3):
rX = fl(S,O)X
1 -rs =-'J1(s,o)x
Yxis
with Equ. 5.169 for J1(s, 0):
S ° fl(S,O) = J1m.x Ks + S Ko + °
The parameters are flm.x, K s, Ko, Yx1s, Yx1o , kLl a, and 0*.
(II.20)
(II.21)
(11.22)
(II.23)
10. "Multiinhibition model" for the quantification of secondary metabolite productions (cr. Sect. 5.4):
Jl(s, 0) = Jlmax s(1 + s/KISX) + K S'x/(1 + p/KIP) Ko + 0 (11.28)
s qp(s,o) = qP,max Kp + s(1 + s/KISP) (11.29)
Combining Equs. 11.4 and 11.29 yielded Equ. 5.138 (KISP == KR == KI!) according to an approach of Bajpaj and Reuss (1981), while the full model of "multiinhibitions" according to Moser and Schneider (1988) contains four constants Kp, KIP, Kisp (== KR), and Klsx ' The main aim of this complex model was to achieve plots of increasing qp at decreasing Jl, as illustrated in Figs. 11.23 and 11.24. The corresponding successful plots of concentrations are shown in Figs. 11.22 and 11.24.
The behavior of the product formation model including such multiinhibition kinetics (with repression), basically represented in Fig. 5.47, is shown in sensitivity analysis in the qp/Jl plots of Fig. 11.25 through 11.29, with variations in all four parameters Kp, KIP' KISP' and KISX'
11. Evaluation of substrate feeding strategy for a fed-batch culture with optimal production of secondary metabolites-case study. On the basis of the mathematical model (Equs. 11.24-11.29) and experimental estimation of model parameters, an optimal S feed can be found by simulations adding the expression Fs(t) for S feed to Equ. 11.25:
F Fs(t) = -'SF
Vo (11.30)
where SF is the substrate concentration [g ,1-1 ] in the feedstream and Fs is the substrate feed rate [gil' h], while F is the volumetric flow rate [1. h -1] and Vo [1] is the initial volume. Figure 11.30 illustrates the concentration/time curves for substrate, biomass, and product at varied S feed F /Vo at constant SF' The corresponding time curves of specific rates Jl in respect to qp show that choice of the right F /Vo allows the value for Jl to be maintained at a minimum while qp is kept constant at a high level (Fig. 11.31).
12. Biosorption model (cf. Sect. 5.3.9). A simple reaction scheme (see Fig. 5.40) is written as a sequence of elimination and degradation
SL elim) Ss + 0 degrad) p (11.31)
The balance equations in case of a DCSTR are (cf. Equs. 5.112-5.115):
434 II. Computer Simulations
(II.32)
(II.33)
and the biosorption rate rads is
(II.34)
According to the literature (Theophilou et ai., 1979; see Sect. 5.3.9), a sorption capacity ~ads is defined (see Equ. 5.115) and used in an analogy to Langmuir adsorption (see Equ. 5.114). The kinetic constants were elaborated by these authors to be load dependent:
ke• = (0.58' fi)x kdegr = (1.86' L - 0.37)x for L < 0.9 h-1
kdegr = (-0.23' L + 1.53)x for L > 0.9
and
Se. = 41· fi + 12 [mg COD '1-1 ]
(11.35)
(11.36a)
(1I.36b)
(II.37)
"Sludge loading" s (= adsorbed concentration Sads) was found to be L dependent
s = 36.5' L + 22 [mg COD '1-1 ] (11.38)
and sads,max == smax = 173 mg .1-1 in this situation. Figure 11.32 represents a typical plot of the time curves of concentrations (Sad., Se., and Sd == Sdegr)'
13. Monod kinetics in CSTR (cf. Sect. 6.1.1). The following set of balance equations were used (see Equ. 6.3):
rx = jl(s)x - D . x
1 - rs = y,- jl(s)x + D(so - s)
xis
(II.39)
(II.40)
with Monod-type kinetics according to Equ. 11.4. Whereas Fig. 6.1 showed the typical plot of "chemostat" behavior (steady-state concentrations x and "8 versus dilution rate D), Fig. 11.33 represents a similar plot with variations in initial substrate concentration So. The "washout state" is indicated (1 x, 1"8) together with a second stationary state characterized by a nonvanishing biomass concentration 2 X, 2"8.
Whereas for batch cultures the final concentrations depend on the initial concentrations, in an open reactor like the CSTR "equifinality" is established, where the end value of stable concentration x is independent of initial values, as shown in Fig. 11.34 in case of jlmax = 0.8 h-1, Ks = 10 mg '1-1, YxlS = 0.5, So = g .1-1 (from ZIMET, Jena).
II. Computer Simulations 435
CSTR behavior in case of more complex kinetics has been shown in Figs. 6.4 through 6.11 and 6.14 through 6.20. Alternative bioreactor operations (CPFR, NCSTR, RR, etc.) were represented in Sects. 6.4 through 6.8.
BIBLIOGRAPHY
General: Ropke, H., Riemann, J. (1969). Analogcomputer in Chemie und Biologie, Berlin:
Springer-Verlag. Knorre, W.A. (1971). Analogcomputer in Biologie und Medizin. Jena: VEB G. Fischer. Romanovsky, J.M., Stepanova, N.V., and Chernavsky, D.D. (1974). Kinetische
Modelle in der Biophysik. Jena: VEB G. Fischer. Levin, S. (ed.) (1981). Modeles Mathematiques en Biologie, Berlin: Springer-Verlag. Knorre, W.A. (1980). Kinetische Modelle in der Mikrobiologie in "Biophysikalische
Grundlagen der Medizin (Beier W., Rosen R., eds.) Stuttgart: Fischer. Spain, J.D. (1984). Basic Microcomputer Models in Biology, Addison-Wesley Pub!.
Comp., Reading, Massachusetts, USA. Rogers, P.L. (1976). In Advances Biochem. Engng. 4,125.
Special: Bajpaj, R.K., Reuss M. (1981). Biotechnol. Bioengng., 23, 717. Moser, A., Schneider, H. (1988). Bioprocess Engineering 3, in press. Theophilou, J., Wolgbauer, 0., and Moser, F. (1979). Gas-Wasser-Fach-Wasser/
Abwasser 120, 119.
APPENDIX III
Microkinetics: Derivation of Kinetic Rate Equations from Mechanisms
The objective of this appendix is to demonstrate the concepts of the rds (rate-determining step) and qss (quasi-steady-state). These are both of great importance in kinetic modeling, as explained in Sects. 204, 4.2, and 5.2. At the same time, some well-known approaches to the microkinetics of enzyme reactions are presented here; these are more fully discussed in sect. 5.2.2.1.
This is the general form of the Langrnuir-Hinshelwood equation applicable to heterogeneous chemical catalysis.
IlL5 Reversible Michaelis-Menten Type
Reaction mechanism:
E+S~{ES}~E+P k-, k-2
(111.22)
The rds concept yields
dp rtot = dt = k+2·{es} - k_2·e·p (111.23)
The qss concept gives
{ } (k+l . S + L2p)eO es = -----''----'''---=----
k+l . S + Ll + L 2P + k+2 (111.24)
Thus
(111.25)
and in equilibrium, where
k+l . k+2 . S = Ll . L2 . P
and using the definition of Keq (cf. Equ. 5.22) resulted in Equ. 5.21. Simplifications are achieved if P = 0 and s = 0 for the initial rates of forward
and backward reaction, which are used in enzyme kinetic studies.
IlL6 Reversible Langmuir-Hinshelwood Approach
Reaction mechanism:
E + s ~{ES} 2±.h{EP} ~E + P k_, ~ ~
(III.26)
The rds concept states that
rtot = k+2{es} - L2{ep} (III.27)
and qss concepts are needed for both complexes {ES} and {EP}:
In analogy to the preceding case (cf. Equs. 111.18 and II1.19) expressions for both complexes are written as
440 III. Microkinetics: Derivation of Kinetic Rate Equations from Mechanisms
eo ·s· Ks {es} - ----'----"--
1+Ks's+Kp 'p
eO'p'Kp {ep} - ---=--=---=--
1+Ks's+Kp 'p
and Equ. III.27 can be rewritten by using Equ. III.29a and b as
( k+2'S'Ks L 2'P'Kp ) rtot = eo -
1+Ks's+Kp 'p 1-Ks's+Kp 'p
(III.29a)
(III.29b)
(III.30)
resulting in an expression already shown in Equ. 5.23. Very often, the resulting rate equation has the sameJorm as Equ. 5.21 or Equ. 5.23 (both of them can be brought in a similar form) but with a different interpretation of the kinetic parameters. There are many mechanisms more complicated than those described here that nonetheless generate the same type of formal kinetics, for example, Equs. 2.54, 5.21, or 5.23.
Index
A absolute deviation, 89 absolute rate, 19,20 absorption, 221 activated state, 203, 436ff activation, 199 activation energy, 199, 200 activity function of product formation,
synchronous/synchronized pulsed and phased culture, 379
systematic approach, 5, 15, 41
T tanks in series model, 77 Teissier kinetics, 217, 232 temperature, 198 temperature dependence, 198 test of pseudohomogeneity, 146ff test system of biolog. test system theory of activated complexes, 203 thermodynamic efficiency, 31, 32, 33 Thermodynamics, 25 Thiele-modulus, 149, 172, 176, 178,
182, 187, 286 Thiele-modulus based on biofilm thick-