Chemical Engineering Science 56 (2001) 3311–3314
www.elsevier.nl/locate/ces
Shorter Communication
A sensitivity approach to manipulated variable selection for control
of a continuous recombinant fermentation
P. R. Patnaik ∗
Institute of Microbial Technology, Sector 39-A, Chandigarh 160 036, India
Received 4 May 2000; received in revised form 6 January 2001
1. Introduction
The choice of the in-ow variable (i.e. a variable associated with the feed stream) to be used as the manipulated variable for the control of a continuous fermentation
employing genetically modi/ed (recombinant) microorganisms is usually between the -ow rate and the concentration of a key component. The former is expressed as
the dilution rate, which is the -ow rate divided by the
(constant) volume of material in the bioreactor, and the
latter is the primary carbon source or a lumped equivalent
for complex substrates.
Although many studies (Bastin & Dochain, 1990; Chidambaram, 1993; Dahhou, Roux, & Chamilothoris, 1992;
Henson & Seborg, 1992) have chosen either of these
variables for di8erent fermentations, there appears to be
neither a consensus on which variable is preferable nor
an easy method to decide without analyzing the bioreactor performance, following a disturbance, with control
based on each candidate variable. This method can be
computationally demanding for complex fermentations,
which often pose sti8 problems. The alternative approach
of a Lagrangian formulation and application of Pontryagin’s maximum principle (Modak, 1993) is not easier.
Since both methods rely on the solution of unsteady state
problems, a steady state approach might o8er an easier
way to choose the manipulated variable. The method proposed here is based on the recognition that the fermentation should be sensitive to the manipulated variable, and
should be able to di8erentiate between recombinant cells
and normal (plasmid-free) cells and between selective
and non-selective media.
∗ Tel.: +91-172-690-223; fax: +91-172-690-585.
E-mail address: [email protected] (P. R. Patnaik).
A recombinant fermentation generates a speci/c protein through cells which contain an externally implanted
plasmid. Cells without this plasmid cannot synthesize
this protein. A fermentation broth normally contains both
kinds of cells. This is because of practical diAculty in
obtaining a broth with only recombinant cells, as well as
theoretical reasons that stipulate that some plasmid-free
cells are essential to sustain a continuous recombinant
fermentation (Hsu, Waltman, & Wolkowicz, 1994).
Since plasmid-free cells consume the substrate but do
not generate the protein, a ‘selection pressure’ is applied to suppress their growth; this may be done either
by adding an antibiotic to which only the recombinant
(plasmid-bearing) cells are resistant or by removing from
the growth medium a component essential for the growth
of plasmid-free cells. In the example considered here,
the latter method was adopted; the sensitivity approach
has been applied to both selective and non-selective
media. However, because selection pressure is usually
applied at the beginning by adding an antibiotic or removing a growth promoter, it does not allow any further
control and is therefore not suitable as a manipulated
variable.
2. Problem denition
Two primary variables of interest in a recombinant fermentation are the concentrations of the
plasmid-containing cells and of cells devoid of the plasmid. The evolution of their concentrations, from a given
initial state, may be described by the equations shown
below. This model is a generalized version (Patnaik,
1999) of that proposed by Ryder and DiBiasio (1984),
and it is valid for many recombinant systems:
0009-2509/01/$ - see front matter ? 2001 Published by Elsevier Science Ltd.
PII: S 0 0 0 9 - 2 5 0 9 ( 0 1 ) 0 0 0 1 2 - 4
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P. R. Patnaik / Chemical Engineering Science 56 (2001) 3311–3314
dS
X + + (S) X − − (S)
= (S0 − S) −
−
;
dt
Y+
Y−
(1)
dX +
= X + [(1 − q)+ (S) − D];
dt
(2)
dX −
= X − [− (S) − D] + qX + + (S):
dt
(3)
dS
B = [+ (S) − − (S)] {D − + (S)}
dD
+
@
− 1 (S0 − S)
+
@D
+
@
@−
+
−
:
− (S0 − S)[ (S) − D]
@D
@D
+ (S) =
m+ S
;
Ks + S
(4)
Expressions for dS=dD; @+ =@D and @− =@D can be
readily obtained from Eqs. (4) – (6), so they are omitted
for brevity.
− (S) =
m− S
:
Ks + S
(5)
3. Application and discussion
The speci/c growth rates may be expressed as:
This system may have one, two or three steady states,
depending on whether or not certain conditions relating
q; D; S0 ; + (S0 ) and − (S0 ) are satis/ed (Hsu et al.,
1994). However, only one of these states is practically
useful because the other two have a zero concentration
of plasmid-bearing cells, and therefore there can be no
synthesis of the recombinant protein. This steady state
can be shown to be (Hsu et al., 1994)
DKs
;
(6)
S=
(1 − q)m+ − D
X+ =
(S0 − S)[D − − (S)]Y +
;
+ (S) − − (S)
(7)
X− =
(S0 − S)[+ (S) − D]Y +
:
[+ (S) − − (S)]Y −
(8)
As mentioned earlier, we focus on X + and X − because
their values determine the rate of production of the protein
of interest. Based on earlier work (Ungureanu et al., 1994;
Vajda & Rabitz, 1992), the sensitivities of X + and X − to
the two manipulated variables, S0 and D, may be derived
from Eqs. (6) – (8) to be:
@X + [D − − (S)]Y +
;
= +
@S0
(S) − − (S)
(9)
AY +
@X +
=
;
@D
[+ (S) − − (S)]2
(10)
[+ (S) − D]Y +
@X −
= +
;
@S0
[ (S) − − (S)]Y −
(11)
BY +
@X −
=
;
@D
[+ (S) − − (S)]2 Y −
(12)
where
dS
A = [+ (S) − − (S)] {− (S) − D}
dD
@−
(S0 − S)
+ 1−
@D
+
@
@−
−
−
;
− (S0 − S)[D − (S)]
@D
@D
As a case study, we consider the production of
-galactosidase by the yeast Saccharomyces cerevisiae
XK1-C2 carrying the plasmid pSXR 125. Its kinetic data
(Cheng, Huang, & Yang, 1997) conform to the model
contained in Eqs. (1) – (5). Glucose was the main carbon
source, with yeast extract as a growth supplement.
Cheng and coworkers applied selection pressure in
an unconventional way. While normally an antibiotic is
added to the fermentation broth to inhibit the growth of
plasmid-free cells, in their experiments the absence of an
amino acid (tryptophan) achieved this objective. Since
plasmid-free cells grew in the presence of tryptophan but
not in its absence, the tryptophan-free medium may be
called selective and the medium containing tryptophan
non-selective. The values of the metabolic parameters for
this strain are listed in Table 1.
For ease of comparison, the sensitivities in Eqs.
(9) – (12) were non-dimensionalized as follows:
1 @X +
;
(13)
s+ = +
Y @S0
+ @X +
D+ = m +
;
(14)
S0 Y @D
Y − @X −
s− = +
;
(15)
Y @S0
+ Y − @X −
D− = m +
:
(16)
S0 Y @D
Table 1
Metabolic parameters for the growth of Saccharomyces cerevisiae
XK1-C2 harboring the plasmid pSXR 125 (Cheng et al., 1997)
Parameter
m+
m−
Ks
q
Y+
Y−
Units
h−1
h−1
g l−1
—
g g−1
g g−1
Value
Selective
medium
Non-selective
medium
0.24
0.10
0.10
0.08
0.26
0.23
0.24
0.265
0.10
0.08
0.263
0.275
P. R. Patnaik / Chemical Engineering Science 56 (2001) 3311–3314
Fig. 1. Sensitivity surfaces for plasmid-containing cells with respect
to glucose concentration in the feed stream. The light surfaces here
and in Figs. 2 and 3 are for a non-selective medium and the dark
surfaces are for a selective medium.
Fig. 2. Sensitivity surfaces for plasmid-containing cells with respect
to dilution rate.
Fig. 1 shows the sensitivity surfaces for plasmid-bearing
cells with respect to glucose concentration in the feed
stream, referred to also as the feed concentration. The
pro/les are -at and horizontal, and the sensitivities in
a selective medium (dark gray) are smaller than in a
non-selective medium (light gray). The surfaces for
plasmid-free cells have not been shown because they
were similar to those in Fig. 1.
Sensitivities with respect to the dilution rate are radically di8erent, both from those in Fig. 1 and between the
plasmid-bearing and plasmid-free cells. As the dilution
rate increases, the sensitivities of recombinant cells in
both media increase in magnitude (note that the sensitivities are negative in Fig. 2), and the sensitivities in a
3313
Fig. 3. Sensitivity surfaces for plasmid-free cells with respect to
dilution rate.
non-selective medium still remain larger than those in a
selective medium. While each pair of surfaces in Figs. 1
and 2 is of similar shape, those for cells devoid of the plasmid (Fig. 3) are di8erent. They are of opposite curvatures
for selective and non-selective media, and depart further away from each other as the dilution rate increases.
Note that this divergence is a consequence of the model
[Eqs. (4) – (8)] and does not depend on the kind of
selection pressure. Therefore, similar results may be expected for any fermentation that has saturating growth
rates. Since the Monodic form expressed by Eqs. (4) and
(5) is applicable to a number of fermentations (Blanch
& Clark, 1996; Patnaik, 1999), so is the sensitivity
method.
While the sensitivity surfaces in Figs. 2 and 3 get increasingly separated as the dilution rate increases, they do
not change signi/cantly with feed concentration. Moreover, the steepnesses of the surfaces increase as the dilution rate approaches the washout value (i.e. all cells are
washed out of the bioreactor, and therefore S =S0 ). Since
large dilution rates are preferred in order to maximize cell
mass productivity, in spite of the actual cell concentrations being low (Blanch & Clark, 1996), the large magnitudes of the sensitivities allow small manipulations in D
to maintain peak productivity without loss of cells. These
observations and the invariant sensitivities with respect
to the feed concentration (Fig. 1) suggest that the dilution
rate is a more e8ective variable to manipulate for bioreactor control. The present method is easier to apply than
those employed earlier since it is based on steady states
and thus requires less computation e8ort. Although an
example with negative selection pressure has been considered, sensitivity studies with positive selection pressure (Patnaik, 1995) also indicate the suitability of this
method.
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P. R. Patnaik / Chemical Engineering Science 56 (2001) 3311–3314
The steady state sensitivities have important implications for bioreactor control. The cell density, X , following a control action may be expressed by a Taylor
series around nominal values, D̂ and Ŝ 0 , of the manipulated variables:
@X @X TD
+
TS0 ; (17)
X (D; S0 ) = X (D̂; Ŝ 0 ) +
@D D̂
@S0 Sˆ0
where the derivatives are the sensitivities according
to Eqs. (9) – (12). If a sensitivity is small, a large
manipulation is required to restore the desired state, which
is biologically detrimental. This limitation of the feed
concentration, S0 , is also supported by control analyses
(Henson & Seborg, 1992; Shimizu, 1993).
Henson and Seborg (1992) observed that the choice
of S0 as the manipulated variable resulted in undesirably
large control actions for both state-space linearization
and input–output linearization. Although D was unsatisfactory for state-space linearization, it provided excellent
regulatory performance in input–output linearizing control. This inference is implicit in the work of Dahhou
et al. (1992), who could control biomass growth in the
presence of noise by an adaptive regulatory controller
which manipulated the dilution rate within an admissible range. These advantages for continuous fermentation
carry over to fed-batch operation, where control of the
in-ow rate maximizes the /nal concentration of product
but manipulation of S0 results in suboptimal singular control (Modak, 1993).
Notation
D
Ks
q
S
S0
t
X+
X−
Y+
Y−
dilution rate, h−1
Monod constant, g l−1
plasmid loss probability, dimensionless
concentration of main carbon source in the bioreactor, g l−1
concentration of main carbon source in the feed
stream, g l−1
time, h
concentration of plasmid-containing cells, g l−1
concentration of cells without plasmid, g l−1
yield of plasmid-containing cells per unit consumption of S; g g−1
yield of plasmid-free cells per unit consumption
of S; g g−1
Greek letters
+
speci/c growth rate of plasmid-containing cells,
h−1
−
m+
m−
D+
s+
D−
s−
speci/c growth rate of cells without plasmid, h−1
maximum possible value of + ; h−1
maximum possible value of − ; h−1
sensitivity of X + to D
sensitivity of X + to S0
sensitivity of X − to D
sensitivity of X − to S0
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