Biopracerr Engineering 1) (1996)
115-219
g Springer-Vlrlag
1996
Start-up sensitivity to the initial state of a batch bioreactor
for a recombinant Escherichia coli in a complex medium
P.R. Patnaik
Abstract The sensitivities with respect to the initial state of
five key variables describing the performance of a batch
bioreactor have been computed from an experimentally
' 'ated kinetic model. The system has a recombinant
.erichia coli strain containing the plasmid pBR Eco gap,
which codes for glyceraldehyde-3-phosphatedehydrogenase
(GAPDH) in a complex medium. Since previous studies have
shown the start-up sensitivities to be particuiarly important,
the initial 10% of the duration of fermentation was chosen as
the time span. The sensitivities of the cell mass, GAPDH and
acetate increased with time while those of glucose and yeast
extract remained practically constant.
Acetate has a crucial role as it functions as both a product
and a reactant. With no acetate in the inoculum, the
sensitivities of acetate increased an order of magnitude faster
than other sensitivities. However, upon addition of acetate
through theinoculum, its sensitivities decreased the fastest and
stabilised beyond a starting concentration of about 1 g/l
whereas other sensitivities stabilised after 5 to 6 gll of initial
acetate. A three-dimensional envelope in the space of acetate
concentration-time-relative sensitivity shows a locus of
concentrations for minimum time-dependent acetate sensitivity; this may be maintained through fed-batch operation.
affinity constant for acetate
affinity constant for glucose
coefficient of maintenance in acetate
m*
maximum value of m A
rnh"l,
coefficient of maintenance in glucose
rnG
llh
maximum value of mG
rnG,
llh
n
empirical constant
PIPo
P
P
Ulml GAPDH concentration at any time
Po
Ulml initial GAPDH concentration
sensitivity of y ; to yj(0) for A,=c
sc(i,j) t
h
time
X
x/xo
X
g/l
cell mass concentration at any time
initial cell mass concentration
xo
g/'
X
YI
Y2
g
. . of symbols
a
-
A
A,
gll
gll
e
-
E
gll
g/l
Eo
g
-
kj
gll
gll
gll
k,G
gll
kt
kz
llh
llh
G
Go
A/Ao
initial concentration at any time
initial acetate concentration
EIEo
yeast extract concentration at any time
initial yeast extract concentration
GIGO
glucose concentration at any time
initial glucose concentration
inhibition constant for acetate-dependent
growth during the acetate phase
inhibition constant for acetate-dependent
growth during the glucose phase
rate constant for acetate phase
rate constant for glucose phase
Received: 22lanuary 1996
P.R. Patnaik
lnstituteofMicrobialTechnolagy,Sector39-A,Chandigarh - 160036, India
KA
KO
gli
gll
llh
-
Y3
Y4
Y3
-
-
a
e
-
P
y:"
g/g
y:A
g/g
y,c
g/g
y:,
g/g
y:,
gig
; x
UIg
Yh
'Jig
Greek letters
no
yield coefficient for cell mass per unit mass of
acetate during acetate phase
yield coefficient for cell mass per unit mass of
acetate during glucose phase
yield coefficient for cell mass per unit mass of
glucose
yield coefficient for yeast extract per unit cell
mass during acetate phase
yield coefficient for yeast extract per unit cell
mass during glucose phase
yield coefficient for GAPDH per unit cell mass
during acetate phase
yield coefficient for GAPDH per unit ceumass
during glucose phase
proportionality constant for plasmid loss
probability
maximum rate of plasmid replicationsaturation constant of the host component of
plasmid replication
regulation function (0 or 1)
regulation function (0 or 1)
exponent of growth inhibition term for
acetate during the acetate phase
exponent of growth inhibition term for
acetate during the glucose phase
Biopraccrr Engineering 15 (1996)
--
216
llh
llh
llh
Ilh
specific growth rate during acetate phase
maximum value of /I*
specific growth rate during glucose phase
maximum value of /I'
ratio of sensitivities, s,(i, j)lso(i.j)
nondimensional time, tp:
gal K (ATCC 23724) by introducing the plasmid pBR Eco gap,
which contains the gene coding for glyceraldehyde-3phosphate dehydrogenase (GAPDH). The modified strain
is resistant to ampicillin whereas the parent strain is
j
not; this accounts for the use of ampicillin in the production
T
medium.
Batch fermentation experiments [I61 were performed in
250 ml Erlenmeyer flasks and 2 liter fermenters, at DH 7 and
37°C. Two media were used: (i) LB with bacteriopeptone
Introduction
(10 gll), yeast extract (5 gll) and NaCl (5 gll), and (ii)
Many useful enzymes, proteins, antibiotics, etc. are now
a
complex medium which contained peptone (10 gll), yeast
derived from microorganisms which have been genetically
extract
(5 gll), NaCl(5 gll), glucose (5-20 gll), KH2P0, (1 gll),
modified by incorporating into the cells a plasmid containing
, K,HPO, (4 gll), FeSO.. 7H,O (0.2 gll) and ampicillin
a specific gene that codes for a desired product. The host cells
(100 mgll). Nancib et al. [I61 have explained that the LB
provide an epvironment conducive to replication of the
medium was used for storage and reactivation of the strain,
plasmid and product formation but they cannot themselves
while all fermentations were carried out in the complex
synthesise the product. Insulin, growth hormones, strepmedium. Experimental details and assay procedures are
tokinase, 0-lactam and tryptophan are some examples of
available in their studies [15, 161.
r-DNA based products [I]. While genetic manipulation yields
a potent organism, its practical use in a bioreactor poses some
3
problems: structural and segregational instability of the
plasmid-bearing cells [2, 31, the effect of fluid mixing in large Mathematical development
reactors [4, 51, the likelihood of contamination [6], and
sensitivity of the fermentation process to disturbances [7].
3.1
Uncertainty in process measurements and the intrusion of Kinetics
disturbances are an undesirable but ubiquitous feature of
Nancib et al. [16] based their kinetic model on two key
bioreactor operation. Being a closed system, batch fermenta- components, glucose and acetate, in the production medium.
tion is less prone to disturbances than fed-batch and
Initially, when little acetate and sufficient glucose are available,
continuous fermentation; it is thus a useful device to study
glucose is utilised preferentially and GAPDH and acetate are
kinetics, mixing effects and sensitivity to perturbations in the the main products. During this 'glucose phase' there is
initial state. It is also preferred for commercial production
marginal consumption of acetate but, as glucose gets exhauswhere large residence times andlor stringent sterility are
ted, acetate becomes the primary carbon source and the
required [I].
organism continues to grow and produce GAPDH. The
Sensitivity analysis provides information on how small
utilisation of acetate requires yeast extract [15], so this also
changes in certain variables and parameters affect the
enters the fermentation model.
performance of a reaction system. Although well established in
For sensitivity analysis it is useful to convert their
other areas, sensitivity analysis of biological reactions, notably mathematical model into dimensionless form. The mass
fermentation processes, is relatively recent. Applications
balances may then be expressed as follows.
include glucose and insulin variations in diabetic patients 181 Cell mass:
and fermentations for penicillin G [9], tryptophan synthetase
[7] and streptokinase [lo].
Most sensitivity analyses deal with physiological or hydrodynamic parameters. Much less attention has been given to
disturbances in the initial condition even though they have
a significant influence on the course of both biological [7, 111
and nonbiological [12, 131 reactions. This is particularly
important in batch and fed-batch fermentations, which are
Glucose:
inherently time-dependent. Previous analyses 17, 9, 141 have
shown that the start-up phase has a major influence on the
subsequent course of fermentation. The present study
therefore addresses the problem of the sensitivity of the
start-up phase to disturbances in the initial state of a batch
Acetate:
bioreactor. To understand an industrially realistic situation,
a complex medium is considered instead of a convenient
synthetic medium.
/[*
pi
PG
G
I[",
2
Experimental background
This study is based on an Escherichia coli C600 gal K (GAPDH)
strain. Nancib et al. [15] derived the strain from E. coli C600
(3)
P.R. Patnrik: Start-up sensitivity la the initial s t a t e d a batch bioreactor
Yeast extract:
s(i, j ) =?yjlZyj(0).
de
ykxoxg
-=dr
(KGIGo+g){l+(a~dk~)"}E,
Normal disturbances in the initial state y(0) are small
enough to permit linearisation around this state [191. Then the
temporal evolution of the sensitivities is obtained as the
solution of the equations [18]:
-
4~y,&/d&xa
( K , I A ~ + ~ )+(aAolk~)*}E0pf,'
{~
(4)
ds
'
d,'fy[~(o), U,V]>
GAPDH:
(12)
(13)
with the initial condition:
These equations incorporate the specific growth rates of
plasmid-bearing cells in the glucose and acetate phases,
.sed through semi-empirical inhibition kinetics:
e
f, is the Jacobian matrix evaluated at the initial state.
Equation (13) was solved for different starting concentrations of acetate. To indicate each such case, let s,(i, j ) denote
the value of s(i,j) at any instant of time for an initial acetate
concentration of c. The sensitivities for different values of c are
compared with the corresponding values (i.e. at the same time
T) for zero acetate concentration in the starting culture by
defining the ratio:
ac(i,j)=s<(i, j)lso(i, j).
The maintenance coefficients, mC and mA,decrease hyperbolically with increasing growth rate [16] and may be written
as:
r n G = m i l ( l +pGlkf,),
(8)
mA=m:',l(l +pAIk2).
(9)
The regulation functions 4 and y are introduced in Eqs. (1)
to (5) to account for the switching of cell growth from
giucose-dependence to acetate-dependence. Since acetate is
utilised only in the presence of yeast extract 1151:
=I;
;::,
The plasmid loss probability, a, follows the relation 1171:
where p + = p A +pG is the total specific growth rate of
plasmid-containing cells.
3.2
Sensitivity
The sensitivity analysis employed here is based on the direct
differential method [13, 181. Equations (1) to (5) may be
written compactly as:
where y is the vector of state variables, usually concentrations,
and u and v contain the manipulatedvariables and parameters
respectively. The sensitivity of yi to the initial value of y, is
defined as:
(14)
For our system y = [x g a epIT, SO Eq. (13) is a Set of 25
differential equations; these were solved by the fourth order
semi-implicit Runge-Kutta method.
4
Results and distusslon
For this study the initial state chosen for the bioreactor was
representative of those used by Nancib et al. [16]:
X0=0.5 gll;
GO= 19.Ogll; Eo=5.0gll.
Values of all parameters were taken from their Table 1.
Acetate has a special role in the reaction system because
initially it is a product but later participates also as a reactant.
Therefore its effect on the sensitivities was determined by
using different starting concentrations, Ap, in the range 0 to
10 gll so as to include the values of 6 g/l at which the specific
growth rate of plasmid-containing cells is maximum 1161.
Although no GAPDH is present in the beginning, a small value
of Po= 1 Ulml was used in order to avoid numerical difficulties;
this has no significant effect on the results because the GAPDH
activity rises rapidly to 200 Ulml.
A time span of 0 < r < 1 was chosen, which corresponds to
1 hour of real time. This is about 8 to 10% of the duration of
fermentation and is sufficiently short to permit the linearisation contained in Eq. (13). In Figs. 1 to 4 the sensitivities with
respect to only the cell mass, x, are shown because the
dimensionless sensitivities of each of the variables x, g, a, e,
andp with respect to itself and the other variables turnedout to
be equal although, of course, they differed among the varfables.
This seems a serendipitous consequence of nondimensionalisation because it reduces each set of five sensitivities to
just one.
Figs. 1 and 2 show the variation of the sensitivities with time.
In Fig. 1 the initialacetateconcentration is zero (set to 0.001 g/l
in the computer program so that AIAo does not become
infinite). The salient features of Fig. 1 are: (i) sensitivities of
glucose and yeast extract do not vary whereas the others
Fig. 1. Variation of sensitivities with time for a starting culture with no
acetate
Fig. 3. Profiles of the sensitivities of cultures with different initial concentrations o f acetate relative to their values for cuitures with no acetate
initially
Fig. 2. Temporalvariation of the ratios of sensitivities far starting cultures
with 10 gll and 0 gll of acetate
increase with time, and (ii) the sensitivity of acetate is an order
of magnitude larger than the others. Although acetate
sensitivities are large when no acetate is present in the
inoculum, the addition of some acetate reduces these
sensitivities much more rapidly than its effect on the
sensitivities of other variables. This is illustrated in Fig. 2,
where the ratio of sensitivities with 10 g/l and 0 g/l of acetate
(at r=O) have been plotted. Figs. 1 and 2 together imply the
existence of some optimal concentration of acetate in the
starting culture to achieve minimum sensitivity. A similar
optimum was also observed by Nancib et al. 1161 for the
specific growth rate, which was maximum with an initial
acetate concentration of 6 g/l. The low sensitivities of the
primary substrate (glucose) and of the recombinant product
(GAPDH)have also been observed in E. coli strains harboring
other plasmids [7, 141.
Fig.4. Effect of time and acetate concentration on the deviations of the
sensitivities of acetate from their corresponding values (i.e. at the same r )
for cultures with no initial acetate
The effect of acetate, which functions as a reactive product,
on the sensitivities was studied by varying the starting
concentration of acetate in the experimental range [15, 161.
Fig. 3 presents the results at 7=0.2. In conformity with Fig. 2,
the introduction of acetate reduces the sensitivities. The
optimal concentration, however, depends on the particular
sensitivity of interest. All sensitivities stabilise beyond a certain
concentration but that of acetate reaches a plateau earlier
(at about 1 d l ) than those of the cell mass and GAPDH (both
at 5 to 6 dl); as expected, the sensitivities of glucose and yeast
P.R. Patnaik: Start-up sensitivity to the initial state of a batch bioreactor
extract are sensibly independent of acetate. Conservatively,
therefore, an initial acetate concentration of 5 to 6 g/l may be
considered adequate to obtain constant sensitivities, but this is
discussed further in the next paragraph. Since most of the
dec:ease in the sensitivity of acetate occurs in a narrow range
ofits concentration, a magnified view of this range (0 to 0.1 gll)
has been shown in the inset of Fig. 3; it maybe seen that even as
little as 0.05 g/l of acetate in the starting culture reduces the
sensitivity to about a third of the corresponding value without
acetate.
These results suggest that an optimal time-varying concentration of acetate might generate a dynamic minimal sensitivity
profile. To ascertain this, the case of no initial dosage of
acetate was used as a datum and deviations from the sensitivities
for this case, i.e. deviations from Fig. 1, were computed as
functions of acetate concentration and time. The resulting
*'.---e-dimensionalenvelope is shown in Fig. 4. The ridge at the
:orresponds to the maximum sensitivities and obviously it
should be avoided. Each curve also has a minimum, for
a particular combination of acetate concentration and time. In
order to maintain low sensitivities at all times, the acetate
concentration in the fermenter should follow the locus of these
minima. This implies fed-batch operation with acetate being
monitored and supplied continually. Such a time-varying feed
policy is also supported by a similar analysis of continuous
culture 1201, where an optimal dynamic dilution rate was
derived. It might be argued that, because acetate acts as an
inhibitor, its concentration should always be kept low so as to
help the production of GAPDH. However, this also makes the
fermentation more sensitive to process disturbances. Such
a conflict between two desirable objectives has also been
reported before [14, 201; those results and the present study
indicate that 'optimal' operation in recombinant fermentation
may have to compromise between high productivity and low
sensitivity.
Conclusions
This study has analysed the sensitivities of five key components of a complex medium to perturbations in their initial
concentrations during batch fermentation with a recombinant
E. coli strain producing GAPDH. The nondimensional
sensitivities of glucose and yeast extract vary little with time
while those of the cell mass, acetate and GAPDH increase.
Although the sensitivities of acetate are an order of magnitude
larger than other sensitivities for an acetate-free starting
culture, the addition of acetate at the time of inoculation
reduces these sensitivities fastest. Most of this decrease occurs
within 0.1 gll of initial acetate, and the acetate sensitivities
stabilise at much lower starting concentrations (about 1 g/l)
than do the other sensitivities (at 5 to 6 gll). An acetate
concentration of 5 to 6 gll was also reported [16] to maximise
the specific growth rate, and thus appears to be a good choice
from both sensitivity and growth considerations.
This inference was expanded by studying a three-dimensional envelope of the sensitivities of acetate, which shows
a locus of acetate concentration to be followed during
fermentation so as to maintain minimum sensitivity at all
times. This may be possible through fed-batch operation with
acetate being monitored and supplied continually. Such an
inference is supported by a previous study [20] of continuous
fermentation with this strain, where a time-varying optimal
dilution rate was derived.
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