Nonlinear model predictive control of fed-batch ...

Nonlinear model predictive control of fed-batch cultures of E. coli: performance and robustness analysis L. O. Santos ∗ L. Dewasme ∗∗ A. Vande Wouwer ∗∗ ∗ on

leave from the Department of Chemical Engineering, Faculty of Sciences and Technology, University of Coimbra, Portugal ∗∗ Service d’Automatique, Universit´ e de Mons (UMONS), Boulevard Dolez 31, B-7000 Mons, Belgium (e-mails : Laurent.Dewasme, [email protected])

Abstract: This work addresses the control of a lab-scale fed-batch culture of Escherichia coli with a nonlinear model predictive controller (NMPC) to determine the optimal feed flow rate of substrate. The objective function is formulated in terms of the kinetics of the main metabolic pathways, and aims at maximizing glucose oxidation, while minimizing glucose fermentation. As bioprocess models are usually uncertain, a robust formulation of the NMPC scheme is proposed using monotonicity arguments and a min-max optimization problem. The potentials of this approach are demonstrated in simulation using a Monte-Carlo analysis. Keywords: predictive control, min-max optimization, overflow metabolism, fermentation, biotechnology. 1. INTRODUCTION Industrial vaccine production is usually achieved using fedbatch cultures of genetically modified yeast or bacteria strains, which can express different kinds of recombinant proteins. From an operational point of view, it is necessary to determine an optimal feeding strategy (i.e. the time evolution of the input flow rate to the fed-batch culture) in order to maximize productivity. The main problem encountered comes from the metabolic changes of such strains in presence of feeding overflow. This ”overflow metabolism”, also called ”short-term Crabtree effect”, is a metabolic phenomenon that is induced when the rate of glycolysis exceeds a critical value, leading to a generally inhibiting by-product formation from pyruvate. It occurs for instance in S. cerevisiae cultures with aerobic ethanol formation, in P. pastoris with aerobic methanol formation, in E. coli cultures with aerobic acetate formation or in mammalian cell cultures with the aerobic lactate formation. To avoid this undesirable effect, a closed-loop optimizing strategy is required, which could take various forms including nonlinear closedloop strategies based on predictive control [Pomerleau, 1990, Chen et al., 1995, Akesson, 1999, Dewasme et al., 2007, Hafidi, 2008]. Nonlinear model predictive control (NMPC) is needed especially for nonlinear, unsteady batch processes where a trajectory needs to be followed from the prediction of a nonlinear model. It is especially useful for processes operating at or near singular points (e.g., sign changes in process gain and input multiplicities) that cannot be captured by linear controllers and where higher order information is needed. It is necessary for processes with wide swings in operation, beyond the ranges of a local linearization. These characteristics are observed in

many process problems including changeovers in continuous processes, tracking problems in startup and batch processes and the control of nonlinear reactors. For these processes NMPC uses the nonlinear dynamic model to predict the effect of sequences of control steps on the controlled variables. There is a vast and rich literature with overviews on NMPC developments, research, and applications (e.g., Qin and Badgwell [2003],Cannon [2004]). Some of these works address the problem of robust nonlinear model predictive control of batch processes (e.g. Nagy and Braatz [2003, 2004]). An overview of more recent developments on NMPC can be found in Magni et al. [2009] and references therein. This study considers the control problem of a lab-scale fedbatch culture of Escherichia coli with a nonlinear model predictive controller to determine the optimal feed flow rate of substrate. The objective function is formulated in terms of the kinetics of the main metabolic pathways, and aims at maximizing glucose oxidation, while minimizing glucose fermentation. As bioprocess models are usually uncertain, a robust formulation of the NMPC scheme is proposed using monotonicity arguments and a min-max optimization problem. The potentials of this approach are demonstrated in simulation using a MonteCarlo analysis. The paper is organized as follows. A general dynamic model formulation for this class of biosystems is introduced in Section 2.1 along with a presentation on the type of operation regimes the cultures can be subject to, i.e., respiro-fermentative or respirative regime. This regime characterization problem is further analyzed in Section 2.2 to define the control objective function for the bioreactor system. Section 3 provides a brief description on the nonlinear model predictive formulation to control the bioreactor as well as simulation results for the perfect model case. Finally, the robust NMPC problem with

respect to uncertainty on a subset of three kinetic parameters is presented in Section 3.1. Final remarks and future work directions are pointed out in Section 4. 2. MODEL AND CONTROL OBJECTIVES 2.1 Bioreactor model The mathematical model to describe the dynamics of the fedbatch culture of E. coli is a generic model that would, in principle, allow the representation of the culture of different strains presenting an overflow metabolism (yeasts, bacteria, animal cells, etc). The cell catabolism in this biosystem is described through three main reactions, the substrate (glucose) oxidation (1a), the overflow reaction (fermentation of glucose) (1b), and the metabolite product (acetate) oxidation (1c): ϕ1

(1a)

ϕ2

(1b)

k1 S + k5 O −→ X + k8 C k2 S + k6 O −→ X + k3 A + k9 C ϕ3

k4 A + k7 O −→ X + k10 C (1c) In these mass-balance equations, X, S, A, O and C are respectively the biomass, substrate (glucose), acetate, oxygen, and carbon dioxide. These symbols are also used to denote the corresponding concentrations in the mathematical formulation along the paper. ki are the yield coefficients and ϕi are the nonlinear growth rates given by: ϕi = µi X , (2) where µi , i = 1, · · · , 3, are the specific growth rates dependent on the operating regime [Rocha, 2003, Hafidi, 2008]: µ1 = min(qS , qS,crit )/k1 (3a) µ2 = max(0, qS − qS,crit )/k2 (3b)  max(0, qA )/k4 if qS kOS + qA kOA 6 qO ,
µ3 = (3c) max 0, (qO − qS kOS )/kOA /k4 otherwise. The kinetic terms associated with the substrate consumption qS , the critical substrate consumption qS,crit (generally dependent on the cells oxidative or respiratory capacity qO ), and the product oxidative rate qA are given by: S qS = qS,max (4a) S + KS qO,max O qO KiO = (4b) qS,crit = kOS kOS O + KO KiO + A A KiA qA = qA,max (4c) A + KA KiA + A These expressions take the classical form of Monod laws where qS,max , qO,max and qA,max are the maximal values of specific growth rates, KS , KO and KA are the saturation constants of the corresponding element, and KiA and KiO are the inhibition constants. This kinetic model is based on Sonnleitner’s bottleneck assumption [Sonnleitner and K¨appeli, 1986] which was applied to a yeast strain Saccharomyces cerevisiae (Figure 1). During a culture, the cells are likely to change their metabolism because of their limited respiratory capacity. When the substrate is in excess, i.e. S > Sscrit , the cells produce a metabolite product P through fermentation, and the culture is said in respirofermentative (RF) regime. On the other hand, when the substrate becomes limiting (S < Scrit ), the available substrate (typically glucose), and possibly the metabolite P (as a substitute

Fig. 1. Illustration of Sonnleitner’s bottleneck assumption for cells limited respiratory capacity. carbon source), if present in the culture medium, are oxidized. The culture is then said in respirative (R) regime. Component-wise mass balances to the fed-batch bioreactor give the following differential equations: dX = (µ1 + µ2 + µ3 ) X − D X (5a) dt dS = −(k1 µ1 + k2 µ2 ) X + D Sin − D S (5b) dt dA = (k3 µ2 − k4 µ3 ) X − D A (5c) dt dO = −(k5 µ1 + k6 µ2 + k7 µ3 ) X − D O + OTR (5d) dt dC = (k8 µ1 + k9 µ2 + k10 µ3 ) X − DC −CTR (5e) dt dV = Fin (5f) dt where Sin is the substrate concentration in the feed, Fin is the inlet feed rate, V is the culture medium volume and D is the dilution rate, D = Fin /V . OTR and CTR represent respectively the oxygen transfer rate from the gas to the liquid phase and the carbon dioxide transfer rate from the liquid to the gas phase. Classical models of OTR and CTR are given by: OTR = kL a(Osat − O) (6a) CTR = kL a(C −Csat ) (6b) where kL a is the volumetric transfer coefficient and, Osat and Csat are respectively the dissolved oxygen and carbon dioxide concentrations at saturation. Further details on the model formulation can be found in [Rocha, 2003, Hafidi, 2008]. 2.2 Control objectives First, we show that the respiratory capacity has an influence on the critical substrate concentration level. In the optimal operating conditions (S = Scrit ), the fermentation and metabolite product oxidation rates are equal to zero and the substrate consumption rate qS is equal to qS,crit or qO /kOS . Consequently, after a trivial mathematical manipulation of (4a), a relation between the critical substrate concentration level and the cell respiratory capacity is obtained as: KS qO (7) Scrit = kOS qS,max − qO Figure 2 shows a plot of this relation where the point [0, 0] corresponds to a totally inhibited respiratory capacity, preventing any growth, and the point [qO,max , Scrit,max ] corresponds to maximum productivity (i.e. absence of metabolite product (acetate) in the culture medium and a sufficient level of oxygenation). Obviously, the presence of the product in the culture medium

0.6

0.04 0.5

(qO

0.035

, Scrit

max

) max

reaction rates

0.03

Scrit [g/l]

0.025 0.02

0.4

ϕ1

0.3

ϕ2 ϕ3

0.2

ϕ1 − ϕ2 0.015

0.1

0.01

0 0

0.005 0

0.05

0.1

0.15

0.2

0.25

0.3

concentration of substrate, S (g/L) 0

1 qO [g/g/s]

2 −4

x 10

Fig. 2. Scrit as a function of qO . can decrease the respiratory capacity and in turn the value of the critical substrate concentration S = Scrit . In order to maintain the system at the edge between the respirative and respirofermentative regimes, it would be necessary to determine online the critical substrate concentration (Scrit ) and to control the substrate concentration in the culture medium around this value [Dewasme et al., 2009]. Unfortunately, the substrate concentration measurement is a difficult task as typical concentration levels are below the resolution of currently available probes (or sensors). An alternative solution is to reformulate the problem not as a maximization of the respiratory capacity but as the maximization of the substrate consumption rate coupled to the minimization of the fermentation rate. This can be formulated as follows [Dewasme et al., 2009]: max Y = max(ϕ1 − ϕ2 ) (8) Scrit

Scrit

where Y is the assumed measurable cost function, and ϕ1 and ϕ2 correspond to the reaction rates µ1 X and µ2 X, respectively. In order to estimate the cost function Y online, we use a pseudosteady state assumption. Indeed, assuming that the variations of substrate, oxygen and carbon dioxide concentrations are equal to zero, we obtain from (5b), (5d) and (5e): D(Sin − S) = (k1 µ1 + k2 µ2 ) X (9a) −DO + OTR = (k5 µ1 + k6 µ2 + k7 µ3 ) X (9b) DC +CTR = (k8 µ1 + k9 µ2 + k10 µ3 ) X (9c) Dilution terms can be considered as negligible in comparison with OTR, CTR and DSin . Replacing the reaction rates µi X by ϕi , i = 1, 2, 3, (9) can be written: DSin = k1 ϕ1 + k2 ϕ2 (10a) OTR = k5 ϕ1 + k6 ϕ2 + k7 ϕ3 (10b) CTR = k8 ϕ1 + k9 ϕ2 + k10 ϕ3 (10c) From this on, after some basic mathematical manipulations, it is possible to express a relation evolving proportionally to ϕ1 − ϕ2 , as a function of the yield coefficients, OTR, CTR and DSin . We denote DSin as the ”substrate intake rate” (SIR) and obtain: Y = ϕ1 − ϕ2 ∝ y (11) with y = (k1 + k2 ) k10 OTR − (k1 + k2 ) k7 CTR + (k7 k9 − k5 k10 + k8 k7 − k6 k10 ) SIR (12)

Fig. 3. Reaction rates and optimization criteria as a function of the concentration of substrate, S. This optimization criteria can thus be evaluated on the basis of 3 measurements (OTR, CTR and SIR) coupled to a sufficiently good identification of several yield coefficients. Figure 3 shows the evolution of the reaction rates and of the optimization criteria (11) as a function of the substrate concentration for the model of E. coli where the respiratory capacity is assumed to be constant (no oxygen limitation and no inhibition). 3. NONLINEAR MODEL PREDICTIVE CONTROL A nonlinear model predictive control (NMPC) strategy is applied to maximize the production of biomass. A straightforward manner to define the objective cost is to consider the total biomass over the prediction horizon. Alternatively, we can consider an objective cost to track the metabolite byproduct concentration at a small value of reference as in Hafidi [2008], Dewasme et al. [2007]. As pointed out in Section 2.2 these quantities are not practical and/or are difficult to measure online. Defining the objective function in terms of (12) overcomes these issues because it can be evaluated on the basis of 3 measurements. To test this conceptual approach in maximizing the biomass production over the prediction horizon, here we define the objective function in terms of (11), assuming that ϕ1 and ϕ2 can be estimated/measured, such that p

m

ref Ψ = ∑ (ϕ1,k+i − ϕ2,k+i ) + β ∑ (Fin,k+i−1 − Fin,k+i−1 ) i=1

(13)

i=1

where p is the output predictive horizon, and m is the input control horizon, with m 6 p. Fin is the manipulation variable, Finref is the feed rate of reference, and β is a control penalty constant. The NMPC problem can be stated as max.Ψ (14a) u

s.t. x˙ = f (x, u, d, θ) (14b) u(t) = u(ti−1+m ), t ∈ [ti+m ,ti+p ] (14c) xL 6 x 6 xU (14d) uL 6 u 6 uU (14e) ∆umin 6 ∆u j−1 6 ∆umax , j = 1, · · · , m (14f) where (14b) is the process model, x and u are respectively the vector of state and control variables, d is the vector of input variables, and θ is the vector of parameters. The subscripts L and U stand respectively for lower and upper. In this case study u = {Fin,k , · · · , Fin,k+m−1 } is the feed rate policy over a control horizon of m sampling time intervals, and x is the augmented vector of the state predictions {xk+1 , · · · , xk+p }. Linear inequalities (14f) enforce control move rate constraints over the

g/(g.h) g/L g/L g/(g.h) g/L g/L g/L

kOA KiA k1 k2 k3 k4

2.0 3.17 250. 1.996 5.850 3.164 25.22 10.90 6.382

g/L L g/L g/L g/L g/g g/g g/g g/g

control horizon. In this application, it is assumed that oxygen is non limiting and has no significant influence on the bacterial growth. Therefore, in this case study the nonlinear model (14b) is simply defined by (5a,5b,5c,5f), and is integrated over p T = [X to obtain xk+i k+i , Sk+i , Ak+i ,Vk+i ], i = 1, · · · , p. Figure 4 shows the closed loop profiles of the state variables and of the controller actuation with predictive horizons (p, m) = (6, 3). The sampling-time is of 6 min, and constraints are defined such that x > 0, V 6 4.85 L, 0 6 Fin 6 0.4 L/h, and ∆umin = −0.005 L/h and ∆umax = 0.005 L/h. These profiles are obtained with β = 2.5 × 102 in (13). β was tuned by trial and error and was needed in order to prevent excessive control moves away from the reference control proref , · · · , F ref file, {Fin,k in,k+m−1 }. The initial profile of the feed rate of reference is constant and equal to the initial feed rate (Table 1). Then at every sampling-time this profile is updated with the optimal feed rate trajectory from the NMPC problem solved at the previous time instant. In each NMPC problem the candidate solution for the feed rate policy over the control horizon m is initialized with a profile equal to the reference control profile. The constrained nonlinear multivariable problem (13 – 14f) is solved numerically with Matlab using the Successive Quadratic Programming (SQP) solver fmincon and the ODE solver ode15s. In fmincon the SQP problem is solved using the active-set algorithm, and the gradients of the objective function are estimated with finite differences. As for the integration of the model equations, the Jacobian of the system is approximated by finite differences in ode15s. With this configuration the computational effort is of approximately 3 minutes to simulate the NMPC application for a bioreactor operation time of 20 hours on a 2.49 GHz Pentium(R) Dual-Core computer. We observe in Figure 4 the profiles for the perfect model case, with the typical exponential biomass growth over a bioreactor operation time of 20 hours. The initial state conditions, inputs, and the model parameters are given in Table 1. The biomass at t = 20 h is 110.6 g with a corresponding biomass productivity of 0.0158 g/(h · g of glucose). The integral of the mass of acetate over the entire time of operation is of 85.8 g h. This value provides a valuable indicator on the degree of exposure of the culture medium to acetate during the 20 hours of operation. 3.1 Robust NMPC As bioprocess models are always uncertain, especially with regards to the kinetics, the issue of robustness of a NMPC scheme is of fundamental importance for practical applicability. In this section, a robust formulation of the predictive controller is proposed based on the solution at each sampling time of a

X (g/L)

1.832 0.1428 2.020 0.7218 6.952 0.09670 0.5236

S V Sin

30 25 20 15 10 5 0 0

5

10

15

20

0

5

10

15

20

0

5

10

15

20

0

5

10

15

2.0

S (g/L)

qS,max KS kOS qO,max KiO qA,max KA

g/L g/L L/h

1.5 1.0 0.5 0.0 2.0

A (g/L)

5.0 0.5 0.001

1.5 1.0 0.5 0.0 5.0 4.5

V (L)

X A Fin

4.0 3.5 3.0 20 Fin ref Fin

0.2

Fin (L/h)

Table 1. Initial state conditions, inputs, and model parameters [Hafidi, 2008, Rocha, 2003].

0.1

0.0 0

5

10

15

20

t (h)

Fig. 4. Perfect model case. Closed loop evolution of the states, X, S, A, and V , and feed flow rate Fin and Finref with predictive horizons (p, m) = (6, 3) and ∆t = 6 min. minimax problem: min. max{Ψ1 , · · · , Ψη } u

s.t.

(14b – 14f)

(15a) (15b)

where each cost function, Ψi , i = 1, · · · , η, defined as in (13), is evaluated at one of the vertices of the uncertainty parameter polytope. The consideration of the polytope vertices only is justified by the monotonicity of the model equations, and by the resulting reduction in computational load. To illustrate this approach, we consider an uncertainty of ±15 % around the nominal values (Table 1) of three kinetic parameters: qS,max , qO,max , and qA,max . Therefore, the uncertainty space can be described by a polytope defined by eight vertices. The problem (15) is solved for the largest value of a set of η = 8 cost functions. To show the performance of the proposed approach, the robust NMPC problem is solved through a series of Monte Carlo (MC) experiments with uniformly distributed random uncertainty on those three plant parameters. Each MC experiment requires the solution of problem (15) at every sampling-time (∆t = 6 min) for a total time of operation of 20 hours. The computational effort is of 1.8 hours per MC experiment. Therefore, although it would be worthy to perform these experiments with a larger set of parameters, that could be achieved only at the expense of a considerable computational effort. The simulation results described here are obtained with the same control problem settings as described before for the perfect model case (Figure 4).

25 20

N

X (g/L)

30 25 20 15 10 5 0

10 0

5

10

15

5

20

2.0

S (g/L)

15

0 1.6

1.5

1.7

1.8

qS,max

1.9

2.0

2.1

1.0

25

0.5

20 0

5

10

15

20

N

0.0

A (g/L)

2.0

15 10

1.5

5

1.0

0

0.5

0.65

0.70

qO,max

0.0 0

5

10

15

20

5.0

0.80

25

4.5

20

4.0

N

V (L)

0.75

3.5

15 10

3.0 0

5

10

15

5

20

0 0.085

0.090

Fin (L/h)

0.2

0.095

0.100

qA,max

0.105

0.110

Fig. 6. Histograms of the plant parameters qS,max , qO,max , and qA,max during the MC experiments. N stands for number of MC experiments.

0.1

0.0 0

5

10

15

20

180

t (h)

160

Fig. 5. Plant/model mismatch case. Closed loop evolution of the state variables X, S, A, and V . Profiles of the 520 Monte Carlo experiments.

4. CONCLUSION The high productivity of fed-batch cultures using genetically modified strains exhibiting overflow metabolism relies on a

120

N

100 80 60 40 20 0 60

70

80

90

100

Biomass production (g) 180 160 140 120 100

N

A total of 520 MC experiments were made. Figure 6 shows the histograms of the plant parameters qS,max , qO,max , and qA,max during the MC experiments. The corresponding closed-loop profiles are represented in Figure 5. The penalty parameter β in (13) to prevent excessive control moves away from the reference control profile as well as the control move rate constraints (14f) lead to a more restricted feed rate policy. As a consequence the acetate concentration is driven to lower values than the ones observed in Figure 4. The acetate concentration at the end of the bioreactor operation (t = 20 h) is zero in circa of 95 % of the Monte Carlo experiments. The histograms of the obtained final biomass production and productivity, final volume, and integral of acetate mass are given in Figure 7 and 8, respectively. The biomass production is lower than the production in the perfect model case (110.6 g), and it is above 90 g in circa of 76 % of the MC experiments (Figure 7). It is noteworthy to point out that the acetate concentration goes to zero in almost all the experiments, and that the degree of exposure of the culture medium to acetate is quite small in comparison with the perfect model case. The integral of the mass of acetate over the entire time of operation is always smaller than the value obtained in the perfect model case (85.8 g h), and it is inferior to 25 g h in approximately 84 % of the MC experiments (Figure 8).

140

80 60 40 20 0 0.01585

0.01590

0.01595

0.01600

Productivity of biomass (g/(h · g of substrate))

Fig. 7. Plant/model mismatch case. Histograms of the biomass production and productivity. N stands for number of MC experiments. double condition: an optimal feeding strategy and the implied limitation of the inhibiting by-product formation. This is the case of the fed-batch culture of Escherichia coli problem described in this work. A nonlinear model predictive controller was implemented to determine the optimal feed flow rate of

180 160 140 120

N

100 80 60 40 20 0 4.0

4.1

4.2

4.3

4.4

Vt=20h (L) 180 160 140 120

N

100 80 60 40 20 0 0

10

20

30

40

50

60

70

80

90

Integral of acetate mass (g · h)

Fig. 8. Plant/model mismatch case. Histograms of the final volume (Vt=20 h ) and of integral of acetate mass. N stands for number of MC experiments. substrate. The resulting control strategy efficiently regulates the feed rate to simultaneously maximize the biomass production and maintain the acetate concentration at low levels. A robustness study regarding the uncertainty on a subset of three parameters shows the capabilities of the control strategy to handle successfully parameter plant/model mismatch. As future work, this analysis will be extended to a larger set of kinetic parameters and stoichiometric coefficients as well. 5. ACKNOWLEDGMENT This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The first author acknowledges financial support from Fundac¸a˜ o para a Ciˆencia e a Tecnologia, Portugal, under grant SFRH/BSAB/943/2009. The scientific responsibility remains with its authors. REFERENCES M. Akesson. Probing control of glucose feeding in Escherichia coli cultivations. PhD thesis, Lund Institute of Technology, 1999. M. Cannon. Efficient nonlinear model predicitive control algorithms. Annual Reviews in Control, 28(2):229 – 237, 2004. L. Chen, G. Bastin, and V. van Breusegem. A case study of adaptive nonlinear regulation of fed-batch biological reactors. Automatica, 31(1):55–65, 1995. L. Dewasme, F. Renard, and A. Vande Wouwer. Experimental investigations of a robust control strategy applied to cultures of S. cerevisiae. ECC 2007 , Kos, Greece, July 2007.

L. Dewasme, A. Vande Wouwer, B. Srinivasan, and M. Perrier. Adaptive extremum-seeking control of fed-batch cultures of micro-organisms exhibiting overflow metabolism. In Proceedings of the International Symposium on Advanced Control of Chemical Processes - ADCHEM 2009, Istambul, Turkey, 2009. G. Hafidi. Application de la commande pr´edictive non-lin´eaire a` la commande de culture de bact´eries Escherichia coli. PhD thesis, Sup´elec, France, 2008. L. Magni, D. M. Raimondo, and F. Allg¨ower, editors. Nonlinear Model Predictive Control – Towards New Challenging Applications, volume 384 of Lecture Notes in Control and Information Sciences. Springer, 2009. Z. Nagy and R. D. Braatz. Robust nonlinear model predictive control of batch processes. AIChE Journal, 249(7):1776 – 1786, 2003. Z. K. Nagy and R. D. Braatz. Open-loop and closed-loop robust optimal control of batch processes using distributional and worst-case analysis. Journal of Process Control, 14(4):411 – 422, 2004. Y. Pomerleau. Mod´elisation et commande d’un proc´ed´e fedbatch de culture des levures pain. PhD thesis, D´epartement de g´enie chimique. Ecole Polytechnique de Montr´eal., 1990. S. J. Qin and T. A. Badgwell. A survey of industrial model predictive control technology. Control Engineering Practice, 11:733 – 764, 2003. I. Rocha. Model-based strategies for computer-aided operation of a recombinant E. coli fermentation. PhD thesis, University of Minho, Portugal, 2003. B. Sonnleitner and O. K¨appeli. Growth of Saccharomyces cerevisiae is controlled by its limited respiratory capacity : Formulation and verification of a hypothesis. Biotechnol. Bioeng., 28:927–937, 1986.