Proceedings of the 391b IEEE
`
`Conference on Decision and Control
`Sydney, Australia • December, 2000
`
`A simplified probing controller for glucose feeding in
`Escherichia coli cultivations
`
`Mats Akesson*
`
`Per Hagander
`
`Department of Automatic Control, Lund Institute of Technology, Lund, Sweden
`
`Stirrer
`
`Abstract A strategy for control of the glucose feed
`rate in cultivations of E. coli is discussed. By making
`probing pulses in the feed rate it is possible to
`detect and avoid a characteristic saturation linked to
`undesirable by-product formation. A simplification of
`a previous algorithm is presented and analyzed.
`
`1. Introduction
`Many proteins are today produced using genetically
`modified microorganisms. Recombinant DNA tech(cid:173)
`nology makes it possible to insert DNA coding for
`a foreign protein into a host organism, thereby cre(cid:173)
`ating a "cell factory" for the protein. The bacterium
`Escherichia coli is a common host. It can be grown to
`high cell densities but a problem is the accumulation
`of the metabolic by-product acetate, which tends to
`inhibit growth and production (6, SJ.
`Formation of acetate occurs under anaerobic condi(cid:173)
`tions but also under aerobic conditions in situations
`with excess carbon source. Accumulation of acetate
`can be reduced by manipulation of strains, media,
`and cultivation conditions. In fed-batch cultures the
`feed rate of the carbon source, typically glucose, can
`be manipulated to restrict acetate formation. Anum(cid:173)
`ber of feeding strategies have been developed (7, llJ,
`however, most of them require considerable process
`knowledge to work well, and the implementation of(cid:173)
`ten relies on specialized and expensive sensors.
`In [1, 2J we presented a probing feeding strategy
`that requires a minimum of process knowledge and is
`implemented using standard sensors. Lab-scale ex(cid:173)
`periments under various operating conditions have
`shown that the method reproducibly works well. In
`this paper we present a simplified controiler which
`yields facilitated analysis, and better performance.
`
`2. Process Description
`E. coli bacteria are cultivated in a stirred bioreac(cid:173)
`tor with a liquid medium containing cells and sub(cid:173)
`strates, see Figure 1. Air is sparged into the liquid in
`
`• Corresponding author. Present address: Biotecnol SA, Tagus(cid:173)
`park, Oeiras, Portugal. e-mail: ma@biotecnol.com
`
`Figure 1 A stirred bioreactor with incoming feed flow.
`
`order to supply the culture with oxygen. Well-mixed
`conditions are obtained through agitation with a stir(cid:173)
`rer and the stirrer speed is also used to adjust the
`oxygen transfer rate. Control loops for temperature,
`pH, and dissolved oxygen ensure that suitable op(cid:173)
`erating conditions are maintained. After an i:O:itial
`batch phase, the main substrate glucose -is fed at a
`growth-limiting rate. In this way, the feed rate can
`be used to manipulate the glucose uptake and the
`growth rate. Typically, the process is divided into a
`growth phase where the cell mass is increased, and
`a production phase where production of the recom(cid:173)
`binant protein is induced.
`We will now briefly recapitulate a model for an
`aerobic fed-batch cultivation of .E. coli. Particular
`attention is paid to the dynamic relation between
`dissolved oxygen and glucose feed rate. For details
`on parameters and metabolic expressions see [lJ.
`
`2.1 Metabolic Relations
`The cell metabolism is described by the specific rates
`of growth f1, glucose uptake qg, oxygen uptake Qo,
`and net production of acetate Qa· The metabolic
`expressions used are largely similar to the ones
`presented in [12J. Glucose provides energy and raw
`material for cell growth. The specific glucose uptake,
`Qg, is taken to be of Monod type
`
`which describes a smoothly saturating glucose up(cid:173)
`take. Oxygen is used to metabolize the glucose, and
`the specific oxygen uptake q0 depends on Qg·
`
`0-7803-6638-7/00$10.00 © 2000 IEEE
`
`4520
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`

`
`ce:ax
`
`-cfamax ___________ ,
`
`,;..crit
`,-'' 'l.g
`
`qg
`
`Figure 2 Relations between specific rates of glucose
`uptake, Qg; oxygen uptake, q0 , and net acetate production,
`Qa· Dashed lines show potential acetate consumption and
`its effect on q0 when acetate is present in the media.
`
`Formation of acetate under aerobic conditions typ(cid:173)
`ically occurs when qg exceeds a critical value if,Jit.
`It has also been observed that q0 reaches an appar(cid:173)
`ent maximum q';'ax at the onset of acetate forma(cid:173)
`tion [4, 9J as illustrated by the solid lines in Fig(cid:173)
`ure 2. Acetate present in the medium may also be
`consumed and used for growth. Its uptake mecha(cid:173)
`nism is likewise modeled to follow Monod kinetics
`
`q~pot = q~max Aj(ka +A)
`
`However, as the consumption requires oxygen and
`glucose is the preferred substrate, it is limited both
`by the available oxidative capacity and the uptake
`mechanism. The dashed lines in Figure 2 outlines
`the maximum potential acetate consumption and
`the resulting effect on q0 • The specific growth rate
`J.L increases with the glucose uptake but with a
`decreased yield above if,Jit. Consumption of acetate
`also contributes to the cell growth.
`
`2.2 Bioreactor Model
`Component-wise mass balances for the bioreactor
`give the following" equations
`
`dV =F
`dt
`d(VX) =
`dt
`d(VG) = FG· -
`dt
`
`(G A)· VX
`'
`
`J.L
`
`(G)· VX
`Qg
`
`Ln
`
`oxygen transfer coefficient, KLa, increases with the
`stirrer speed N but is also affected by factors like
`viscosity, foaming, and anti-foam chemicals.
`When the measurement of dissolved oxygen is used
`for control, it is also important to consider the dy(cid:173)
`namics in the dissolved oxygen probe. In practice,
`most sensors measure the dissolved oxygen tension
`which is related to the dissolved oxygen concentra(cid:173)
`tion through Henry's law 0 = H · C0 • The probe is
`here modeled as a first-order system
`
`(1)
`
`2.3 Linearized Model
`Over short periods of time, V and X are approxi(cid:173)
`mately constant. The dynamics for deviations in glu(cid:173)
`cose uptake and dissolved oxygen due to changes in
`F may then be approximated by the linear equations
`
`(2)
`
`(3)
`
`where the gains and the time constants vary sig(cid:173)
`nificantly during a cultivation. Typically, Tg and To
`decrease with increasing biomass. Below the respira(cid:173)
`tory saturation, equations (2) and {3) together with
`the model for the sensor (1) give a third-order linear
`model relating 11F and flOp In [1J it is shown that
`the stationary gain can be rewritten as
`
`K = KgKo = -(0*- Osp)/({1 + a)F)
`
`(4)
`
`where a normally is close to 0 but may approach 1 for
`high acetate concentrations or low glucose uptakes.
`
`2.4 Control Problem
`It is often desirable to keep F high since this will give
`faster growth and a minimized process time, thereby
`improving productivity. However, to avoid accumu(cid:173)
`lation of acetate the glucose feed rate F should be
`kept low so that overflow metabolism is avoided and
`aerobic conditions maintained. In other words, the
`resulting specific glucose uptake qg should be less
`than ifJit and the oxygen consumption less than the
`maximum achievable oxygen transfer in the reactor.
`The handling of the limited oxygen transfer was de(cid:173)
`scribed in [1J and we will here concentrate on avoid(cid:173)
`ing overflow metabolism. A major challenge is the
`time-varying and uncertain nature of the process.
`For instance, if,Jit is often poorly known and it may
`also change during a cultivation, especially during
`production of a recombinant protein [3, lOJ.
`
`(G A)· VX
`'
`Qa
`
`d(VA) =
`dt
`( *
`d(VCo)
`~ = KLa(N) · V Co- Co)- q0 (G,A) · VX
`
`where V, X, G, A, Co are the liquid volume, the
`cell concentration, the glucose concentration, the ac(cid:173)
`etate concentration, and the dissolved oxygen con(cid:173)
`centration, respectively. Further, F, Gin. c; denote
`the feed rate, the glucose concentration in the feed,
`and the dissolved oxygen concentration in equilib(cid:173)
`rium with the oxygen in gas bubbles. The volumetric
`
`4521
`
`BEQ 1012
`Page 2
`
`

`
`Figure 3 The probing control strategy. By making
`probing pulses in F it is possible to determine if Qg is
`above or below q'f:it from the response in Op.
`
`3. A Probing Control Strategy
`The key idea of the probing feeding controller is to
`exploit the characteristic saturation in the respira(cid:173)
`tion that occurs when qg exceeds ct;it and acetate
`formation starts. The saturation can be detected by
`superimposing probing pulses in F that are long
`enough to be seen through the system dynamics. As
`long as q9 is below cfi/it, the pulses give rise to re(cid:173)
`sponses in the dissolved oxygen signal Op. However,
`when qg is above q'j{it, q0 is saturated and no re(cid:173)
`sponse is seen. In this way it is possible determine
`if qg is above or below ct;it without knowledge of the
`actual value [3]. This information is then used to ad(cid:173)
`just F to achieve feeding at ct;it, see Figure 3. The
`probing controller can be viewed as an extremum
`controller, see [5], controlling the process to a satu(cid:173)
`ration instead of an optimum.
`
`3.1 A Simplified Feedback Algorithm
`The previous implementation of the algorithm [1, 2]
`made use of both up and down pulses in a switching
`scheme depending on the pulse responses. The down
`pulses are here omitted while maintaining, and
`even improving, performance. The simplification also
`makes implementation and analysis easier.
`At each cycle of the algorithm, a pulse is given to
`get information. If the amplitude of the response in
`dissolved oxygen /Opulsel exceeds a threshold level
`Oreac. the feed rate is increased as
`
`t1F(k) = K: · F(k)/Opulse(k)//(0*- Osp)
`
`(5)
`
`Otherwise, if /Opulsel < Oreac, it is concluded that
`q9 exceeds ct;it and F is decreased with a fixed
`amount chosen to be equal to the pulse size Fpulse·
`An example is shown in Figure 4 where also some of
`the algorithm parameters are defined.
`To ensure that Op starts from the same level, Osp.
`at each pulse, it is regulated by manipulation of the
`stirrer speed. During the feed pulses, however, the
`stirrer speed is fixed in order not to interfere with
`the detection algorithm. If disturbances cause Op to
`deviate from the set-point at the time for a pulse,
`a safety net delays the pulse to avoid erroneous
`interpretations of the pulse response.
`
`,_f
`F~:k)
`
`Tpulse
`
`Tcontrol
`
`0 ~ ~-:::--'\ Osp
`
`P ......... v ......... :oreac
`
`Figure 4 Example of a cycle in the algorithm. During
`the time Tcontrol between two pulse!!, Op is regulated by
`a controller manipulating the stirrer speed.
`
`0
`0
`
`0.5
`
`1.5
`
`2.5
`
`3
`
`3.5
`
`A (g/1]
`
`Op (%]
`
`Qg (g/(gh)] ~
`40 ~
`
`0.5
`
`1.5
`
`2.5
`
`3.5
`
`4
`
`0.06~
`Fjljhj 0·:~
`
`0.04
`
`0
`
`0.5
`
`1
`
`2.!;
`
`3
`
`3.5
`
`4
`
`1.5
`2
`Time (h]
`Figure 5 Simulation with K: = 1 where the initial feed
`rate is chosen too low. After two hours q'!fit is changed.
`The dotted lines indicate the critical glucose uptake,
`q'f:it, and the reaction level Oreac. The probing controller
`achieves feeding at q~·it without prior information of the
`value and in spite of the change.
`
`3.2 Simulation Example
`A simulation of the simplified algorithm is shown in
`Figure 5 where the initial feed rate after a batch
`phase gives a specific glucose uptake qg below ct;it.
`The controller increases F until ifJit is reached and
`qg is then kept approximately constant as desired.
`After 2 hours the value of q~rit is decreased, and
`as can be seen the algorithm is able to detect and
`adjust the feed accordingly. The undershoot is due
`to reconsumption of the acetate produced when qg
`temporarily exceeds ct;it. The accompanying oxygen
`consumption gives q0 = q'(/ax even though Qg < cft1
`and hence F is 'decreased as no oxygen responses
`can be seen. This example illustrates the ability to
`achieve feeding at c{grit without a priori information
`and in spite of changes that may occur in the process.
`
`4522
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`BEQ 1012
`Page 3
`
`

`
`\Opulse\
`
`Figure 6 Amplitude of the pulse responses in dissolved
`oxygen \Opulsel as a function of q11 • The shaded area
`indicates the region where \Opulse I < Oreac and q11 < q"J11
`•
`
`3.3 Guidelines for Tuning
`In the next section, we show how to choose the "con(cid:173)
`troller gain" 1C in (5). Tuning rules for the other pa(cid:173)
`rameters were derived in (1] based on the linearized
`process model. The only required information is an
`upper bound for the lumped time constant T max in
`the relation between F and Op together with the
`noise level in Op. These quantities are mainly equip(cid:173)
`ment related and do not depend critically on a par(cid:173)
`ticular strain or product.
`
`4. Stability Analysis
`The closed-loop system is non-linear and contains
`elements of continuous-time as well as discrete-time
`nature and a complete analysis is therefore difficult.
`However, by considering it as a sampled-data system
`it is possible to derive a sufficient condition for
`convergence to a region below but close to cf;it.
`
`4.1 Characterization of Oxygen Responses
`When checking the oxygen response Opulse to pulses
`in the feed rate F, the pulse length Tpulse is chosen
`equal to the estimated process time constant T max
`as recommended in [1]. This is long compared to
`the glucose dynamics, and we may then assume
`qgulse R;j Kg · Fpulse· Consider the response to a pulse
`starting at Qg. Obviously, if Qg is above cfJit no
`response will be seen. For Qg far below cfr!it, the
`amplitude of Opulse is given by the linearized model
`,8\Ko\ · qgutse where 0.63 < ,8 < 1
`as \Opulse\ =
`accounts for the dynamics and the finite pulse time.
`As Qg approaches to cf,/it, the saturation in q0 will
`cause a reduced response, and only the part of the
`pulse that is below cf,{it will contribute to Opulse· This
`is summarized in the piecewise linear function
`if Qg > ct;it
`if Qg<QC:it-~ulse (6)
`
`0,
`
`,8\Ko\~ul~e,
`
`\Opulse\ =
`
`{
`
`,B\Ko\(qc;1t- qg), otherwise
`
`shown in Figure 6. The shaded area indicates where
`\Oputse\ < Oreac even though Qg < cfJu. As Oputse also
`depends on ,8, which changes during a cultivation,
`the width of the shaded region will change.
`
`q _,.max
`c{amax o- "'.o
`,,
`\' na:
`\' ,.
`,, / '
`'
`\'
`,,
`'
`\\
`\ ,,
`\"\
`\'
`\\
`'
`'
`1\ \';
`'
`., ..
`\, ..
`\ ,.
`
`•,\
`
`\ \ I
`\,I
`
`n1
`
`n3
`
`~
`
`Figure 7 Regions in the plane ( q 11, q~.pot). The shaded
`region 02 indicates where \Opulsel < Oreac even though
`q 11 < q"Jit. The arrows indicate qualitative trajectories of
`the closed-loop system.
`
`4.2 Effects from Acetate Consumption
`If Qg < cfr!it, the cells may consume acetate present
`in the media with a concomitant consumption of
`oxygen. It is important to include this in the analysis,
`as the specific oxygen uptake q0 may be saturated
`even if Qg < cf;u. In Figure 7 we introduce the
`plane (qg,q~pot) where q~pot = cfamaxA/(ka +A). To
`the right of ct;lt acetate is produced, and to the left
`acetate may be consumed. In the region between the
`two dashed lines, we have q0 = q':,'ax and hence the
`acetate consumption is there limited by the available
`respiratory capacity. To the left of this region the
`acetate consumption is limited only by- the uptake
`mechanism and q0 < q':,'=. As before, the shaded
`region indicates where \ Opulse\ < Oreac and Qg < cf,;it.
`We will denote this region 0 2 , and the regions to its
`left and right, Q 1 and ila, respectively.
`
`4.3 Closed-loop System
`The controller connected to the process is now
`treated as a sampled-data system with sampling pe(cid:173)
`riod Tpulse + Tcontrol and we consider the possible tra(cid:173)
`jectories in the plane ( Qg, q~pot). Acetate is consumed
`to the left of the line qg =¢/it. Hence A and q~pot will
`decrease in this region. Conversely, when Qg > cf;it,
`acetate is produced and q~pot increases. In the Qg(cid:173)
`direction the controller directly affects the process.
`The choice of the sampling interval assure that the
`glucose dynamics can be neglected and thus
`
`~qg(k) = qg(k + 1)- Qg(k) =Kg· ~F(k)- e
`
`(7)
`
`where e accounts for the influence from the chang(cid:173)
`ing cell mass VX. In Qb we have \Opulse(k)\ > Oreac·
`Hence !:!F(k) > 0 and qg(k) will increase. Con(cid:173)
`versely, in il2 and il3, ~F(k) < 0 and qg(k) will
`decrease until these regions are escaped to the left.
`Qualitative trajectories, when neglecting the drift
`from thee-term, are indicated by the arrows in Fig(cid:173)
`ure 7. Note, however, that the discussion below holds
`even if e is larger than zero.
`
`4523
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`BEQ 1012
`Page 4
`
`

`
`''~
`
`'
`
`0.05
`
`0o
`
`1
`
`2
`
`3
`
`4
`
`c,pot
`Qa
`
`::r t. ~~~ .. uAMJ
`
`0
`
`1
`
`2
`Qg
`Time [h]
`Figure 8 Oscillatory behavior when the feed incre(cid:173)
`ments are too large, K = 2.5. Left: Trajectory of the
`sampled-data system in the plane (qg, q~,pot). Right: Feed
`rate F and acetate concentration A.
`
`3
`
`4
`
`0.1r~
`005~~
`0o
`02n~
`011~
`
`1
`
`2
`
`3
`
`4
`
`0o
`
`1
`
`2
`Qg
`Time [h]
`Figure 9 Convergent behavior when K = 1. Left:
`Trajectory of the sampled-data system in the plane
`(qg,q~,pot). Right: Feed rate F, acetate concentration A.
`
`3
`
`4
`
`We see that if fl.qg(k) is too large, we cannot exclude
`"limit cycles" where the process is oscillating between
`.Q 1 and .Qg. An example of this is shown in Figure 8
`which shows a simulation with ~ = 2.5. The key to
`avoiding oscillatory behavior is to ensure that fl.qg ( k)
`is smaller than the shortest distance from .Q 1 to the
`line if;it. Then all trajectories starting in .Q 1 or .Q2
`will remain to the left of if;it. Furthermore, as all
`trajectories to the left of c{grit tend downwards, we
`can conclude that there exist attractive invariant
`sets M in the lower right part of this region. As all
`trajectories starting to the right of c{grit will end up
`to its left, any M is globally attractive.
`
`4.4 A Sufficient Condition for Stability
`The condition that fl.qg ( k) should be smaller than the
`shortest distance from n 1 to the line if;it will now
`be translated to a sufficient bound on 1C. For any
`\Opulsei > 0, we know from (6) that JOpulse/Koi <
`ifJit- Qg· Now, by using e 2: 0 in (7) together with
`(5) and ( 4) we get
`
`fl.q (k) < K fl.F(k) < 1CJ0pulse(k)J < ~(,..crit _ q )
`-
`g
`g
`-(1+a)JKoi-
`'ig
`g
`(8)
`
`:::; 1. A
`Hence, stability is ensured if we choose 1C
`simulation with 1C = 1 is shown in Figure 9. As
`expected the system converges, without oscillations,
`to the desired region to the left of qc;it.
`
`5. Setpoint and Integral Action
`In the simulation shown in Figure 9, the system
`seems to converge to a well-controlled "stationary"
`situation where the oxygen response is larger than
`Oreac· This is not always so, even if 1C :::; 1, see for
`instance the later part of the simulation in Figure 5.
`A linear analysis reveals conditions for the existence
`of a fix point in the "proportional band" below c{grit,
`and suggests the introduction of a set-point and
`integral action to enhance the convergence.
`
`4524
`
`5.1 Convergence to Stationadty
`Assume that A ~ 0 and that any acetate resulting
`from a probing pulse is consumed before the next
`
`pulse. For convenience, introduc•3 y = I Opulsei· Inside
`
`the proportional band defined by (6), y is given by
`
`From the process equation (7) we can obtain
`y(k + 1) = y(k)- PiKoJKgfl.F(k) + PIKoie
`Using (5) and (4) we finally get
`
`(9)
`
`(10)
`
`y(k + 1) = {1- P~/(1 + a)}y(k) + PIKoie
`(11)
`and when 0 < ~ < 2(1 + a)j p, we see that y(k)
`converges to
`
`y* = (1 + a)JK0 Jej1C
`Hence, if the drift from e is such that y* > Oreac,
`the process will converge to a stationary point below
`c{grit. However, if Oreac is too large or e is too small
`the controller will change between increments and
`decrements as in the right part of Figure 5.
`
`(12)
`
`5.2 Introducing a Setpoint and Integral Action
`By introducing a set-point Yr in the control law
`
`fl.F(k) = 1C. y(k)- Yr . F
`0*- Osp
`
`(13)
`
`the fix point y* is shifted to
`
`Hence, by choosing Yr > Oreac we get y* > Oreac.
`even if e = 0. If Yr is too large, y saturates and
`the maximum control signal gives a Qg below the
`proportional band. A better solution is to introduce
`integral action
`
`fl.F(k) ,;, ( ~-(y(k)-Yr)+~i L)YU)-Yr)) o·!:. 0
`
`k
`
`j=O
`
`sp
`
`BEQ 1012
`Page 5
`
`

`
`
`
`.. ~f(gh~J:~.····· ···................. . ..
`
`A (g/1]
`
`0
`
`o
`
`0.5
`
`1
`
`1.5
`
`2
`
`2.5
`
`3
`
`3.5
`
`4
`
`O,J%J~2H
`
`0.5
`
`1
`
`1.5
`
`2
`
`2.5
`
`3
`
`3.5
`
`4
`
`20o
`0.06~
`F ~/h] o.o4
`0 .02 -
`
`.
`
`2.5
`
`3
`
`3.5
`
`4
`
`0o
`
`o.5
`
`1
`
`1.5
`2
`Time [h]
`Figure 10 Simulation with PI controller (K = 0.96,
`K; = 0.64) where q"Jit is changed after two hours. The
`dotted lines indicate the critical glucose uptake, q"Jit,
`and the reaction level Oreac. Feeding at q"Jit is achieved
`and after the transients the amplitude of the oxygen
`responses reach the set-point Yr = 3 which eliminates
`the "oscillations" seen in Figure 5.
`
`which gives y* = Yr if the stability condition
`
`K>O, Ki>O, 2K+Ki<4(1+a)jf3
`
`{14)
`
`is fulfilled. Thus, convergence is assured by choosing
`Yr in the proportional band and larger than Oreac·
`Outside the proportional band the integral is recal(cid:173)
`culated so that the stability condition induced by {8)
`is fulfilled. In this way the global stability property
`is maintained and wind-up effects are avoided. Fig(cid:173)
`ure 10 shows a simulation with PI control ( K = 0.96,
`Ki = 0.64) using Yr = 3. The setting is the same as in
`Figure 5 and again feeding at if,/it is achieved. How(cid:173)
`ever, after the transients the amplitude of the oxygen
`responses reaches the set-point and this eliminates
`the "oscillations" seen in Figure 5. The fluctuations
`seen are due to noise added in the simulation.
`
`6. Conclusions
`In cultivations of E. coli it is important to avoid ac(cid:173)
`cumulation of the by-product acetate, which is pro(cid:173)
`duced when the specific glucose uptake qg exceeds a
`critical value if,/it. We have discussed an approach
`for control of glucose feeding based on a superim(cid:173)
`posed probing signal in the feed rate. It can be imple(cid:173)
`mented with standard instrumentation and enables
`control of qg at an unknown and time-varying if,/it.
`The primary contribution of this paper is a simpli(cid:173)
`fied algorithm and a stability analysis that gives a
`sufficient condition for convergence.
`
`Acknowledgments
`Financial support from Pharmacia & Upjohn, the
`Swedish National Board for Industrial and Technical
`Development (project 1Nll-97-09517), and the Alf
`Akerman foundation is gratefully acknowledged.
`
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