Bioprocess Engineering 11 (1994) 23 28 9 Springer-Verlag1994
Strategy for the simulation of batch reactors when the enzyme-catalyzed
reaction is accompanied by enzyme deactivation
F. X. Malcata
23
Abstract The balance equations pertaining to the modelling of
batch reactors performing an enzyme-catalyzed reaction in the
presence of enzyme deactivation are developed. The functional
form of the solution for the general situation where both the rate
of the enzyme-catalyzed reaction and the rate of enzyme
deactivation are dependent on the substrate concentration is
obtained, as well as the condition that applies if a maximum
conversion of substrate is sought. Finally, two examples of
practical interest are explored to emphasize the usefulness of the
analysis presented.
List of symbols
Ce
mol/m 3
Q,o
mol/m3
Cs
mol/m 3
Cs,o
mol/m 3
Cs, min mol/m 3
k
S/S
kl
s/s
k2
1/S
K,~
p
q
t
mol/m 3
mol/(m 3s )
mol/m 3
s
Greek symbols
~b
SiS
mol/m 3
mol/m 3
~,
sis
concentration of active enzyme
initial concentration of active enzyme
concentration of substrate
initial concentration of substrate
minimum value for the concentration
of substrate
first order rate constant associated with
conversion of enzyme/substrate
complex into product
first order deactivation constant of
enzyme (or free enzyme)
first order deactivation constant of
enzyme in enzyme/substrate complex
form
Michaelis-Menten constant
time derivative of Cs
auxiliary variable
time elapsed after reactor startup
univariate function expressing the
dependence of the rate of enzyme
deactivation on Cs
dummy variable of integration
dummy variable of integration
univariate function expressing the
dependence of the rate of substrate
depletion on Cs
derivative of ~ with respect to Cs
1
Introduction
Batch reactors used to perform enzyme-catalyzed reactions are
fairly common in the biochemical (and fine chemical) industry.
In order to be able to properly design such reactors a priori and
to properly optimize their utilization a posteriori, efficient
techniques of mathematical modelling are required that take
into account various possibilities for the kinetic mechanisms
pertaining to the enzyme-catalyzed reaction and to the thermal
deactivation of the enzyme. A variety of uni- and multisubstrate
mechanisms for the former have been comprehensively
described in the literature [1], and similar observation holds for
unimolecular mechanisms associated with the latter in the
presence of macroheterogeneity [2] or microheterogeneity [3].
The assumption of substrate-dependent rate expressions for
either the enzyme-catalyzed reaction and the deactivation of the
enzyme has been successfully employed before, e.g. in the
determination of the optimum temperature path in the case
of batch reactors performing enzyme-catalyzed reactions [4].
The combination of some of the aforementioned mechanisms for
the enzyme-catalyzed reaction with the material balances for
various reactor configurations aiming at optimized reactor
designs has also been reported to considerable extent [5-lo].
The existence of constraints in terms of the maximum
conversion of substrate ever attainable when enzyme
deactivation is present was emphasized elsewhere for the case of
CSTR cascades [n-s3]. However, the integration of both the
enzyme-catalyzed and the enzyme deactivation reactions with
the material balances pertaining to the reactor has not deserved
extensive attention so far.
This communication attempts to partially fill this gap by
providing a general mathematical strategy for the simulation of
biochemical reactors operated batchwise when both the enzymecatalyzed reaction and the enzyme deactivation reaction take
place; its applicability is tested via the discussion of two selected
examples.
2
Received 28 June 1993
Theory
The unsteady-state mass balances to enzyme undergoing
deactivation following first order kinetics with respect to the
enzyme, and substrate undergoing depletion also following first
order kinetics with respect to the enzyme in a well-stirred reactor
operated batchwise can be written as:
F. X. Malcata
Escola Superior de Biotecnologia, Universidade Cat61ica Portuguesa, Rua Dr.
Ant6nio Bernardino de Almedia, 4200 Porto, Portugal
dCe
dt
~'
m3/(mol s)
-
O{cAc~,
O)
BioprocessEngineeringn 0994)
with initial condition:
C~=Q,o,
t=O,
(2)
Equation (13) has the form of a general first order ordinary
differential equation, which has the following solution:
/
,4{ ~1
p - - - O { c s } [ CE, 0+ cs,]0cs-4'{- 4} d~) .
and
05)
%
dCs
dt
-
(3)
-O{Cs}C~,
Recalling Eq. (12), one finally observes that:
4{{}
with initial condition
t=O, Cs=Cs, o,
24
respectively, where CE and Cs are the molar concentrations of
active enzyme and substrate, respectively, t is time elapsed since
startup of the reactor, 4 is a univariate expression relating the
rate of enzyme deactivation with substrate concentration, and
O is a univariate expression relating the rate of substrate
depletion with substrate concentration.
Equation (1) may be replaced by the expression obtained by
division of Eq. O) by Eq. (3):
(5)
(16)
which can be rewritten as:
(~{~) (d~
-- d(
Q~
d
t= I
(~7)
Cs
The auxiliary variable q is defined as:
Cs=Cs, o, CE=CE,o 9
(6)
Rewriting Eq. (3) as:
1 dC s
0{ Cs} dt"
(7)
and differentiating Eq. (7) with respect to t, one gets:
O' l d C s \ 2 1 (d2Cs~_
d-~s)t--dT-)--~Tt--&-) +~ t, dt 2 ] - 0 ,
q ~- 6
(18)
Recall Eq. 06), which can be written in the alternative form:
and, therefore, Eq. (2) should be replaced by:
(dCE\/dCs\
'
(-p)
4{ Cs}
dG 0{ c A '
dee
CE=
+fg 4{~}
(4)
cs,0
d~
t=*
(
fCs,o4{~} ) "
Cs 0{(} CE,0-d{
(19)
For t = 0% one will obtain the maximum ever attainable
conversion of substrate, Cs,mln. Since the integration range is
finite, inspection of the denominator of the integrating function
in Eq. (19) indicates that, in such situation, Cs,min should satisfy
the following condition:
(8)
Cso 4{ 4}
r
minI=OVC<o -
5 ~d~=0.
(20)
Cs,mln ~ } g /
where:
The first condition is trivial, since for every type of kinetics for
substrate consumption by an enzyme-catalyzed reaction, the rate
dCs "
of reaction is obviously nil when the concentration of substrate
is zero. The second condition is much more constraining. It
Combination of Eqs. (8) and (5) yields:
arises from the fact that there are two competing factors in the
reacting
system, the deactivation of enzyme, which leads to
2Cs~
(dCs'~ O' fdCs~ 2
(lO)
a
decrease
in the number of active enzyme molecules with time,
=o,
\dt}
O\dt/I
and the rate of the enzyme-catalyzed reaction, which leads to
which now requires two initial conditions, viz. Eq. (4) and
a decrease in the number of substrate molecules with time;
hence, longer reaction times imply more extensive conversion of
des
(11) substrate per unit molecule of active enzyme but also fewer
t=O, T = -O{ Cs,o} CE,o.
molecules of enzyme left active, which may lead to the existence
of a maximum for the conversion of substrate. It should be
Equation (lo) is a second-order, second-degree ordinary
emphasized that Cs,m~n> Oif4 is a weaker function on Cs than is
differential equation which can replace Eq. (3). Solution of
irrespective of the value of CE,0, which is equivalent, in
Eq. (lo) may proceed through the introduction of the new
physical
terms, to saying that 4{Cs}IO{Cs} becomes
dependent variable p, defined as:
unbounded when Cs tends to zero; the equality Cs,~i~ = o can be
des
guaranteed a priori only if the enzyme is actually reactivated by
p ~-~-,
02)
the presence of substrate at whichever concentration, i.e. if
4{ Cs} < o; in the remaining cases, e.g. 4{ Cs} 10{ Cs} equal to
which, combined with Eq. (10), leads to [14] :
a constant or limc,~o 4{ Cs} 10{ Cs} equal to a finite value, the
decision on which solution of Eq. (2o) holds depends on the
dp'~ O'
(13) numerical values of the parameters in question.
The applicability of the foregoing general analysis is explored
(14) below in two examples of practical interest.
G=G,o, P= - 9 { G,o} G,o.
O'- dO
(d-<i7r)+4
(
(9)
F. X. Malcata: Modelling batch enzyme reactors with deactivation
3
First numerical example
Assume that the rate of deactivation is independent of the
concentration of substrate. In this situation, ~b{ Cs} reduces to
a constant, and Eq, (17) becomes:
/
t-
-tr
- -
1
\
(21)
In
1
4qCs,o} i ~
C~,o
c,
(22)
"
Assuming Michaelis-Menten kinetics for the substrate
consumption and first order deactivation of enzyme, the result
is:
k,
ci~ K~ +
(
(23)
d~},
where kl is the first order deactivation constant, K~ is the
Michaelis-Menten parameter i.e. the dissociation constant of the
enzyme/substrate complex, and k is the first order constant
associated with transformation of the enzyme/substrate complex
into product and concomitant regeneration of the free enzyme
form. After simple manipulation of Eq. (23), one finally gets:
k~t= - l n f l
j"
_Km
_ t C1ii
s,
Cs, o
or, using Eqs. (15) and (18):
t=-
Cso k~(Km+~)
k'Cs'~ ( K ~ l n
(Cs, o)
Cs \ ]
1.0
dC~
dCe
k Cs
dt
K m + Cs
CE,
(27)
C~,
(28)
0.8 ~
~
l
~ ~ - - - - ~
~0.6
k'CE'~176
k.CE,o/(k,.Cs,o)=5
oF/
0
I
s
K~ + Cs
kCo,,o o
O m 0.4
k'CE,o/(k1-Cs,o)=I
k-CE,o/(k1.Cs,o)=5~
3
4
5
k 1 .t
1.0
klKm§
dt
1.0
0
2
i
25
Second numerical example
Assume now that the free enzyme deactivates via a first order
irreversible process with constant kl and that the enzyme in the
enzyme/substrate complex form deactivates via a first order
irreversible process characterized by constant k2, which, in
general, will be different from kl; assume in addition that the
substrate consumption follows Michaelis-Menten kinetics as in
the previous example. In this situation, Eqs. (1) and (3) take the
form:
o
1
(26)
4
I
~0.6
0L
0
~kCEo
'
--0.
kl Cs, o
Plots of CS,mi~/Cs, o are available in Fig. 2 as functions of
parameters kCE, o/(klCs, o) and K,dCs, o.
0.8
0.2
o l --Cs'min+l
( Cs, min )
Cs, o
and
Plots of Cs/Cs, o vs. k~ t are available in Figs. la to lC for a number
of values of parameters kC E,o/ ( k ~C s, o) and k~/ C s, o and K J C s, o.
0.4
(25)
d~=C~,o,
which is equivalent to:
q{cs,o} d~/
~ --,
o}) ,~{Cs} U
t
Equation (20) now takes the form:
0
1
i
3
2
,
4
O
5
k1.t
I
0.8
o
,~0.6
0~0.4
0.2
00
" ~ k.CE,o/(k~.Cs,o)= 1
k-CE,o/(k1-CE,o)=5
~ ~ ~ .
1
Fig. la~c. Plots of the dimensionless substrate concentration, Cs/Cs,o, vs.
the dimensionless time, kl t, for (a) K,~/Cs,o = o.1, (b) K,,/Cs.o = 1 and (c)
Km/Cs,o=lo
k.CE,o/(kl.Cs,o)=l0
2
3
k 1.t
4
5
Bioprocess Engineering 11 (1994)
100
26
5
Discussion and conclusions
It is interesting to note that in both examples studied the
minimum conversion of substrate ever attainable is a bivariate
6O
function as apparent from inspection of Eqs. (26) and (33); if one
moves from example 1 to example 2, parameter kCE,ol(ks Cs,o)
("~,- 10 "I
E
should be substituted by the lumped parameter [kC~,o/
(ks Cs, o ) l / ( g J C s , o), whereas parameter Km/Cs,o should be
0
substituted
by parameter k2/ks. Inspection of Fig. 2 indicates
9Km/Cs,o=0.1 Km/Cs,o=5
Km/Cs,o=lO
that the normalized value of ln(Cs, msn) is essentially a linear
function of parameter kCe,o/ (kl Cs,o), with a common vertical
10 2
0
10
20
30
40
50 intercept at unity and a slope that decreases as the dimensionless
value of K m increases. With respect to Fig. 4, it is apparent that
(a) for k 2/k~ = o, in (Cs, min ) is always a decreasing linear function
of kCe, o/(k~gm), and that (b) such linear behavior is also
Fig. 2. Log-lin plot of the dimensionless minimum substrate concentration,
Cs,min/Cs,o, VS. the dimensionless parameter kCe,o/(k~Cs,o) for
observed for other values of kz/kl provided that parameter
KmlCs,o=O,1, KmlCs,o=5, and KJCs, o=m
kQo/(k~g~) takes large values.
Careful inspection of Eq. (31) allows one to conclude that if
ks
=
k2, then example 2 reduces to example 1. This fact should be
respectively, so Eq. (16) becomes:
expected because it implies that the rate of deactivation of
Km+~
enzyme is the same irrespective of the forms it takes, and it is
cs
k - - - ~ d(
known that the relative amounts of either free or conjugated
t=- j
(29) forms of enzyme are direct functions of the concentration of the
,~ k~Km+k2~
Cs,oCE,o + jQo
-k~ " d~
substrate. It should be noted that reactivation of the enzyme by
substrate implies that the intrinsic activity of the enzyme
actually increases when substrate is present, as compared with
Algebraic manipulation of Eq. (24) yields:
the complete absence thereof; this is not the case explored in
example 2 for k21ks<l because in the latter situation the
1
d[
1 Q
k \ v, +k2
presence of substrate leads only to a decrease of the rate of
deactivation of enzyme when substrate exists in large amounts
with respect to its complete absence. The lines depicted in
Figs. 3a to 3f associated with k2/kl = 0 correspond to the
situation where the presence of substrate at any concentration
prevents inactivation of enzyme.
In Figs. i and 3, the lines plotted extend from unity to a certain
+ j
,
(30)
lower asymptotic limit that can be obtained in every case from
Fig. 2 or Fig. 4, respectively. Although in a few of the several
cases depicted in Figs. 1 and 3 it seems that the lines extend
which, upon integration and rearrangement, becomes:
virtually to zero substrate concentration, such is never the case
because the value of Cs, min is actually finite and positive, as
klt= --ln {1--klCs'o (Km In ~Cs.o'~+k2 ( 1 C s "]~+
implied by the fact that q5 is always a weaker function on Cs than
kCE, o \Cs, o ( Cs J kl \
Cs, oJ/IJ "'"
is ~ in all the cases depicted. As expected, when kC~,o/(kiCs, o)
increases, i.e. when either k/k~ or Cs,o/Ce,o increases, the
(
k2) i
617
+ 1-~
k CE 0 Km
k2
(31) minimum concentration of substrate attainable decreases
because the effect of klt on the enhancement of the rate of the
c~/C~,ok~ C~'os+~s.o in{ 17}+ kll ( ' 7 - 1 )
enzyme-catalyzed reaction is stronger than said effect on the
enhancement of the enzyme deactivation reaction. Given any set
Plots of Cs/Cs, o vs. k~ t are available in Figs. 3a to 3f for
of values for kCE,ol (klCs, o) and Km/ Cs, o, both increases of k2lk t
a number of values of parameters k Qo/(k~. Cs,o), k2/k~, and
from
zero to unity, and from unity to a higher value lead to
kmlCs, o.
decreases in Cs, min , thus implying that detrimental effects of the
Equation (zo) takes in this situation the form:
presence of substrate on the activity of the enzyme always
Cs,o k~K~+k2~, d ~ = C ~ o ,
(32) constrain the best performance of the batch reactor.
S
Furthermore, the effect of k2/kl is diluted when K,,/Cs, o and
Cs,min
kCE,ol(klCs, o) are large.
which can be rewritten a s :
The analysis presented in this communication is relevant for
the
simulation of batch reactors that perform enzyme-catalyzed
Km (ln ( Cso ] k2[
Cs~'~\ kCe, o
reactions because it includes the effect of thermal deactivation,
s,o.
--~
a factor that very often places severe constraints on the technical
and commercial feasibility of such reactors; The reasoning was
Plots of Cs, rnin/Cs, o a r e available in Fig. 4 as functions of
developed in such way as to possess the advantage of being
parameters [kC ~,o/(k~Cs, o) ] I(K~lCs, o) and k flkl.
k.CE,o/(k 1.Cs, o)
I~iT--,/klKm)
F. X. Malcata: Modelling batch enzyme reactors with deactivation
1.0
i
i
'
1.0
'
1 0 --I
k2/kl=
-=
~-0.6
0.4
k2/kl=O
0
B
I
0.8
0
0.2
l
d
k2/k 1=5
0.8
T
- -
~-0.6
0
0 ~ 0.4
k /kt=lC
0.2
k2/kl=1
k2/k~=O
k2/kl =1
kJk~=5
0
0
2
3
4
5
k 1.t
k 1 .t
1.0
1.0
k2/k t = 10 k~k~ =5
0.8
0.8
~-0.6
k2/k 1= 1
~-0.6
e
l
k2/k 1= 10
k2/k 1=5
k2/kl=1
0
0
k2/k 1=0
0~0.4
0~0.4
k2/kl = 0
0.2
0.2
b
0
_1
1
0
I
I
2
3
--
-
-
0
1
k 1.t
1.0
[
- -
k~ .t
i
--
o
~0.6
O3
1.0
l
kz/k 1=10
k2/k 1=5
ke/k~=l
k2/k 1=0
0.8
0
0.2
J1
3
4
k2/k ~= 10
k2/k ~=5
kJkl=l
0
0.2
0
i
~0.6
O.4
0
i ....
0.8
0,4
0
5
2
5
O0
k2/k 1=0
1
2
4
k 1.t
k 1.t
Fig. 3a-f. Plots of the dimensionless substrate concentration, CslCs, o, vs.
the dimensionless time, kl t, for: (a) kC~,o/ (k~Cs, o) = t and K~ / Cs,o = o.1; (b)
kCE,o/(k~Cs, o)=l and K,~/Cs, o =l; (c) kC~,o/(k~Cs.o)=t and g~/Cs, o=lO;
(d) kC~,o/(klCs, o)=lo and Km/Cs, o=o.1; (e) kCE,o/(kiCs, o)=lo and
K~/Cs, o=l; and (f) kC~,o/(klCs, o)=to and Km/Cs, o=io
dimensionless, a n d having the final form of the solution of the
s i m u l t a n e o u s material balances to s u b s t r a t e a n d enzyme, viz. Eq.
(19), require evaluation of an integral r a t h e r t h a n solution of
a set of two differential equations, viz. Eqs. (1) a n d (3).
0
o
.-
1~ 1
/k I =20
O
10-2 [-.
0
,
I
5
,
I ........
10
15
, ,
20
25
k.Ce.0l(k 1 , K )
Fig. 4. Log-lin plots of the dimensionless minimum substrate concentration,
Cs, mln / Cs, o, VS, the dimensionless parameter [ kC E,ol (k~ Cs,o ) ] I ( K J Cs, o) for
k2/k~ =o, k2/k1=5, k2/k~=z2, and k~/k~ =20
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