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Optimal Control of Fed-Batch Fermentation with Autoinduction of Metabolite Production B. Tartakovsky? S. Ulitzurt and M.Sheintuch'gt Department of Chemical Engineering and Department of Food Engineering and Biotechnology, Technion-Israel Institute of Technology, Haifa, 32000 Israel
Induction of metabolite production is a common feature of many cultures obtained by recombinant DNA technology. Synthesis of a product in these cultures is triggered by a sharp change in environmental conditions or by autoinduction in which the cells, under conditions of growth limitation, produce a species that initiates the synthesis. If product synthesis is the objective of the fermentation, then the time of induction should be optimized so t h a t enough biomass is produced prior to induction while ample nutrient is left in the broth to sustain product synthesis after induction. In this study, recombinant luminous Escherichia coli has been used to portray an autoinductive culture in batch and fed-batch fermentations. A model describing the cell density and substrate and inducer concentrations has been constructed, and its parameters were estimated. An optimal control strategy with three stages of the culture state (growth, inducer synthesis, and growth+product synthesis) was determined and validated experimentally by optimization of the substrate flow rate in a fed-batch fermentation.
Introduction Induction of metabolite production is a common feature of many cultures obtained by recombinant DNA technology. Synthesis of a product in these cultures is triggered by a steep change in environmental conditions, such as a change in temperature (Whitney et al., 1989) or the addition of inducer (Betenbaugh and Dhurjati, 1990). Some cultures have an autoinduction property: under conditions of growth limitation the cells produce an inducer that initiates the synthesis of a product. If product synthesis is the objective of the fermentation, then the time of induction should be optimized so that enough biomass is produced prior to induction while ample nutrient is left in the broth to sustain product synthesis after it. Optimization of autoinductive processes is aimed at maximizing product formation by changing the initial conditions in a batch fermentor, the feed rate and composition in a fed-batch (semibatch) fermentor, or the dilution rate and feed composition in a continuous fermentor. The optimization procedure can be carried out analytically when a reliable mathematical model is available. Optimization of fermentation processes described by unstructured (x-s-p)models, which account for biomass, substrate, and product concentrations, with Monod and Andrews kinetics has been studied extensively. Optimal operating policies for batch, fed-batch (Modak et al., 1986), and continuous bioreactors have been suggested (Constantinides and Rai, 1974; San and Stephanopoulos, 1989). The objective of these studies was the maintenance of an optimal temperature or pH and optimal substrate levels in a fed-batch process. Processes with recombinant protein production are complex, and their modeling requires the description of cellular processes. A number of mathematical models have been developed t o describe plasmid instability
* Author to whom correspondence should be addressed.
t Department of Chemical Engineering.
* Department of Food Engineering and Biotechnology.
(Mosrati et al., 1993)and induction phenomena (Lee and Bailey, 1984; Togna et al., 1993). These are structured models that account for regulation of the expression/ excretion system and for plasmid loss. Although good agreement with experimental data has been demonstrated, these models are too complex for process optimization. We present here a simple mathematical model for the dynamics of an autoinductive culture and employ it for process optimization and control. The culture employed is a recombinant luminous Escherichia coli. Recent studies of the bacterial luminous system (Ulitzur, 1989) revealed similarity between its regulatory control and the control of intracellular protein synthesis. The simplicity of on-line luminescence measurements makes luminous E. coli a very convenient microorganism for modeling an autoinductive culture. The biochemical pathway of bacterial luminescence involves the mixed function of reduced flavin mononucleotide (FMNH2) and a long chain aliphatic aldehyde (RCHO) activated by luciferase (L)(Baker et al., 1992), according to FMNH,
+
+ RCHO + 0, luciferase FMN
RCOOH
+ H,O + light
The oxidation products are regenerated and recycled enzymatically at the expense of reduced nicotine adenine dinucleotide (NADH) and adenosine triphospate (ATP). Thus, the luminescence level reflects the cellular concentrations of ATP, NADH, and luciferase and is dependent on oxygen concentration, feed composition, temperature, and pH. The synthesis of luciferase in the recombinant strain of E. coli is autoinductive: in batch growth it is repressed when cell density is low, and it increases sharply when the concentration of the autoproduced inducer reaches a critical level (Baker et al., 1992). The inducer can also be supplied directly to the culture. Ulitzur and Kuhn (1988) have shown that the HtpR protein is involved in the regulation of the lumi-
8756-7938/95/3011-0080$09.00/00 1995 American Chemical Society and American Institute of Chemical Engineers
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nescence system in E. coli cells harboring the whole lux system of Vibrio fischeri. Induction of HtpR protein by different stress conditions, such as starvation or heat shock, advanced the onset of the luminescence. In the following model formulation, we used these results to create a simple unstructured model. Although luminescence is a very sensitive indicator of cell metabolic state, under normal growth conditions and when temperature and pH are constant and oxygen is not the limiting substrate, the luminescence mainly reflects intracellular luciferase concentration. As explained above, luminescence appears during the growth phase as the inducer concentration ascends. Subsequently, luminescence declines at the stationary phase, reflecting the decrease in luciferase activity. In the following derivations, we treat intracellular luciferase as the desired product and assume luminescence (L) to be proportional to luciferase activity (e):
L = y@x
(1)
with y = 1. Thus, on-line luminescence measurements are used as a measure of the desired product. Previous work from our group has used bioluminescence for characterizing mass transfer and kinetics in a mixed fermentor (Vashitz et al., 1988), for determining the substrate concentration profile along a hollow fiber reactor (Sheintuch et al., 1992), and for characterizing oxygen diffusion and consumption in a single pellet of immobilized living cells (Greenberg et al., 1994).
Experimental Section Bacterial Strain, Media, Cultivation Conditions, and Instrumentation. The culture used was a recombinant E. coli W3110 (provided by Prof. S. Ulitzur) with a pChvl plasmid carrying the luminescence system of the marine luminous bacterium Vibrio fischeri. The fermentation medium was Luria broth (LB). We refer to a standard LB solution composed of 10 g/L tryptone and 5 g/L yeast extract as a solution of substrate concentration unity (s = 1SUA), and diluted solutions are assigned appropriate substrate values. The solutions were prepared in a ionic buffer (5 g/L NaCl) in order to maintain minimal osmolarity. Growth kinetics experiments were conducted in a series of shake flasks kept in a water bath shaker a t 30 "C. Process control experiments were conducted in a computer-interfaced fermentor (New Brunswick, BioFlo), which was operated a t an air flow rate of 1.0 L/L.min and an agitation speed of 700 rpm. The working volume of the fermentor was 1.2 L. A sensitive Technion-made luminometer, capable of detecting 106-1012quantds as determined by the Hastings and Weber standard (Hastings and Weber, 19631, was employed. Luminescence was expressed as light units (LU), where 1 LU equals lo3 quantds. An external flow cell situated on a recycle stream and attached to a luminometer was employed for on-line luminescence monitoring. Cell density was determined from optical density measurements at 600 nm using the corresponding LB solution (LB = 1 or LB = 0.5) as the standard and as the medium for dilutions. The correlation between dry cell weight and ODeO0was obtained and used to calculate the cell density. The carbon dioxide concentration in the outlet gas stream was monitored by an IR analyzer (Beckman 865). A personal computer (PC 486) was used to collect data and to control a peristaltic pump for nutrient supply. A culture supplemented with 30 pg/L chloramphenicol was grown overnight at 37 "C and used as the inoculum for shake flask cultures, as well as for the controlled
fermentor. A synthetic inducer (N-(3-oxohexanoyl)homoserine lactone) was used in several runs to trigger the luminescence regulatory system. Growth Kinetics. The dependence of growth kinetics on substrate concentration was determined in several experiments with initial nutrient concentrations in the range s = 0.25-2.0 SU/L, which were prepared by dilutions of the standard solution after inoculating each flask with 0.5 g/L cells. Growth rates were determined from optical density measurements, which were sampled in 15 min intervals. Figure l a presents the growth dependence on substrate concentration, as determined from the first hour of growth, assuming that substrate consumption over this period is negligible. Each experimental point represents an average of two or three reproductions. Growth kinetics was found to be independent of the inducer concentration used: experiments with s = 2 SU/L and inducer concentrations of i = 20 and 40 pg/L yielded the same growth rates (data not shown). Kinetics of Luciferase Synthesis. A similar experimental scheme was used to find the dependence of the luciferase synthesis rate on inducer concentration: luminescence measurements were conducted with excess substrate (s = 2 SU/L) and 4-60 pg/L inducer. Analysis of luminescence dynamics after addition of the inducer showed that some time elapsed before the cell changed its metabolism and luminescence appeared. To exclude the influence of this transient, 20 pg/L inducer was added to the inoculum, which was cultivated for 10 min t o receive an induced culture. Only then was the inoculum diluted and the inducer added to each flask to form the desired concentrations. Luminescence was recorded every 4 min, and the calculated specific synthesis rate is plotted in Figure lb. The dependence of the luciferase synthesis rate on substrate concentration was determined with a large excess of the inducer (i = 80 pgL) by varying the substrate concentration (s = 0.25-2.0 SUA) (Figure IC). Other conditions were similar to these in the previous experiments. Kinetics of Inducer Synthesis. Batch reactor studies showed that, without the addition of external inducer, luminescence appeared at the end of the growth phase with the onset of culture starvation. A pulse of substrate, added to a highly luminous starving culture, resulted in a decline in luminescence (see Figure 4). Literature data (Adar et al., 1992; Ulitzur, 1989) confirm these results. We suggest that the inducer is deactivated by growing cells, and a considerable build-up of the inducer in culture broth occurs when the specific growth rate falls below a certain threshold. To determine this threshold value of the specific growth rate, a set of batch fermentations with different initial substrate concentrations (SO = 0.5 SU/L and SO = 1 SUA) was conducted (Figure 2). Luminescence appeared a t 4.5 and 6.5 h for the two substrate concentrations and reached a maximal value within a hour. The maximal luminescence value more than doubled with doubling the initial concentration of substrate. Each experiment was repeated twice and good reproducibility was obtained.
Model Formulation and Parameter Estimation We derive here a simple model that simulates the autoinduction of product synthesis and estimates its parameters. The model accounts for the following processes: (1)cell growth, which is limited by substrate concentration (s); (ii) inducer (i) synthesis and its deactivation by growing cells; and (3)intracellular product
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suhstrutr concentration, SUL
tinw. h
2s hatch pnwss with So=l.O sdL 2s
?
inducer concentrationlnkgk time. h
Figure 2. Batch cultivation dynamics showing biomass con-
centration (0)and luminescence (- * -) measurements with initial substrate concentrations of 0.5 SUA (a) and 1.0 SU/L (b). Model predictions are denoted by solid lines.
control purposes. For a fed-batch culture of volume (u),
to which a substrate stream of concentration sf is added at a rate f, the material balances take the following form:
4
tl
0
i 03
I .s
1
1
2
suhsrrae - concentrtion. s u n I~
Figure 1. Kinetic data showing the dependence of the specific
growth rate on substrate concentration (a) and the kinetic dependence of the specific synthesis rate on inducer concentration (b) or substrate concentration (c). Solid lines represent model predictions.
(e) synthesis, which ia also limited by the substrate concentration and occurs only in the presence of the inducer. The model accounts for all of the experimental observations and is sufficiently simple to be employed for
-dv =f
dt The specific mwth rate is assumed to follow Monod kineti& and 6 be independent of inducer concentration, in agreement with our observations (Figure 1). We also assume that the inducer is synthesized at a constant rate and that it is deactivated at a rate proportional to the growth rate. The nonlinear multiplier 246 i ) , with €4, on the right-hand side of eq 2b is added to avoid a negative concentration of inducer. Note that we can set a = 1without a loss of generality by substituting i/afor i, ri,,-/a for ri,-, and kda for k,. The dynamics of intracellular product concentration (e) is modeled by
+
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The other parameters (&,ax, qmax, ki, ka, YXt8, and YP/,) were determined by minimization of the squared error difference i.e., production is possible only in the presence of the inducer. The decay ( - k d ) term in the product balance is added to account for the decline in activity at the end of the fermentation (Figure 2). Following Whitney et al. (1989), this term was assumed to be proportional to luciferase activity. The product concentration per culture volume, p = e x , follows
* dt
i p ( 0 )= 0 = qx - fU
The substrate consumption rate
accounts for all synthesis processes in the system. The last two terms, which account for consumption due to inducer synthesis and maintenance (ri/YiI, and m,),are assumed to be negligible. Equations 2 portray an autoinductive system: at the < p, and product beginning of the process i(0)= 0, qmax is not synthesized (q = 0). When the specific growth rate falls below the threshold value of ri,", the inducer concentration builds up and product synthesis starts. A pulse of substrate during the product synthesis stage may reverse the solution to a growth stage. To optimize product synthesis in a fed-batch culture, the supply rate should be kept sufficiently small so that growth will not exceed the threshold value and yet sufficiently high to ensure the build-up of cells. We estimate the parameters of the model (eqs 2) in two stages: an initial estimate is obtained from the kinetic studies conducted in a series of shaking flasks (Figure l), and an improved estimate is based on the batch fermentation studies (Figure 2). The saturation constants ksl and ks2 were not reestimated in the latter stage. Nonlinear regression analysis was conducted using MATLAB scientific software. From the growth rate dependence on substrate concentration (Figure la), we estimated the parameters in eq 2a to be pmax= 0.45 h-I and K,1= 0.26 SU/L. The limitation constant for product synthesis then was estimated from the synthesis rate dependence on substrate concentration (Figure IC): K,2 = 0.15 SUL. The specific synthesis rate dependence on the inducer concentration (Figure lb) suggests that hila = 10 p g L . Note that the specific synthesis rate was not extended to saturation in the experiment, so that an improved estimation was obtained on the basis of batch fermentation data. To evaluate ri,", recall that inducer appears after the specific growth rate falls below a threshold value (ri,max).Assuming that luminescence and induction appear simultaneously, we use batch fermentation data (Figure 2) to find qmax = 0.36 h-l; we also assume that E is small: E = The yield coefficient of biomass on substrate was evaluated from batch fermentations as YxIsE 14.3 g/SU, and substrate consumption for luminescence initially was assumed to be 10% of that value, i.e., Yplszz 143 LU/SU. We consider parameters to be the estimates of the ksl, ks2, and qmax sufficiently accurate.
M
Ni
N,
i
j
j
in the batch fermentation data (Figure 2). We employed both runs, differing in their initial substrate concentrations, and the average of two reproductions of each run is shown in the figure. Trial values of the parameters were as given earlier, and a fifth-order Runge-Kutta method was applied to solve the model (eqs 2) on each step of a simplex search algorithm. The weighting coefficients were chosen as y1 = 1 and y2 = 0.8 to normalize measurements of the process variables with respect to their average values; M = 2 is the number of fermentations having different initial conditions, and Ni is the number of measurements in the ith fermentation. This estimation procedure yields pmm = 0.59 h-l, qmax= 16.05 x lo3 LU/(gh), ki = 2.98 p a , k d = 1.13 x lo3 LU/ (gh), Yxts= 14.5 g/SU, and YpIs= 200 LU/SU, and good corroboration with experimental results is obtained (Figure 2). Analysis. The control problem in a fed-batch process, described by the model (eqs 2), can be stated as follows: What is the temporal substrate feed rate profile (ftt))that will maximize product mass (p(t3 u(t3) at a given final time of fermentation (tf)? Thus, the performance index to be maximized is
J = p(tf)dt,) subject to
fmin
I At) I f,,
(4)
We initially consider a simplified problem that lends itself to analytical solution. We assume that (i) the change of broth volume is negligible (duldt = 01, (ii)the decay rate is negligible ( k d = 01, and (iii)k,l = ks2 (note that the estimates of these parameters are rather similar). We can then write q = k+i/(Ki i) (where k 1 = q m d pmar)and use p as the control variable. The simplified problem with u E p is to maximize
+
subject to the dynamics
di - jqmax - u-)x i dt
€ + Z
i(0)= 0
By the Pontryagin principle (Bryson and Ho, 1975), this problem is equivalent to maximizing the Hamiltonian
(7) where Vx and Vi (the adjoint variables) are the solutions of
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84
- -- - aH- - k l U - ki +uxVj--dt ai (k, + i)2 (E
d4J _ _
E
+ i>2
dt
The Hamiltonian is a linear finction of the control variable (u),and therefore the optimization problem is a singular control problem:
Optimal control, u*, can be defined as Q>O
singular In the singular region (4
Q 0);(2) inducer synthesis, p = 0 (Q 0); (3) biomass growth+product synthesis, p = constant (4 = 0). In the third stage of the process, the specific growth rate is constant (u = using),which implies a constant value of the substrate concentration in the culture broth.
Process Control In the previous section, we derived an optimal trajectory p(t) (hence, s(t)) for fed-batch fermentation, with the autoinduction of metabolite production subject to certain simplifying assumptions. Here we experimentally validate the proposed policy. An optimal process consists of stages of biomass growth, inducer synthesis, and product synthesis. As was already explained, a fast addition of substrate will increase the growth rate and lead to the failure of the synthesis stage. To demonstrate this “improper” control, we conducted an experiment starting with one-half of the total substrate mass and added the other half in one pulse after a stable luminescent signal was detected (Figure 4, model predictions are denoted by solid lines). That led to a luminescence maximum (i.e., product activity, broken line in Figure 41, which was 40% lower than that of the batch fermentation with the same substrate mass (Figure 2). We attribute the decline in yield to inducer deactivation by growing cells after feeding. The model predicts this behavior well. The difference between experimental data and model predictions is probably due to some inevitable changes in parameters from one run to another; the experiment with the improper control strategy was conducted also, with substrate addition at 4.8 h, showing the same results. In the derivation of the simplified model, we used the growth rate as a control variable and obtained an estimate of the optimal solution. The rate of substrate consumption, however, is predefined by biological properties of the culture and cannot be changed artificially. We now turn to the solution of the general problem, subject t o technological restrictions. The problem formulation
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SO,
1
A
0
F
by changing At) subject to 0 I At) Ifmm. The final time ( t f ) is specified by the condition of zero product synthesis rate due to substrate exhaustion. The mass balance equations describing the process are given by eqs 2. If the first process stage is batch cultivation, then the starting time of inducer synthesis (tl)depends on SO. The inlet stream At) is assumed to contain substrate at constant concentration sf. The strategy obtained from the solution of the simplified problem (eqs 5 and 6) suggests that s(t) should follow
2
z
. I
i
dt
= -rp
+f
ds - = -rs(s*)3C dt
(Sf
-s)
+f
(Sf
t, 5 t
- s*) = 0 t,
5
t, + d t (18)
+ Qt < t 5 tf
where t z is the beginning of the third process stage and dt is a short interval of feed pulse to attain the optimal substrate concentrations*. In terms of the feed flow rate Mt)),the optimal policy is
,-e
x
I
,,'
[LiE
U*
Sf
- s*
t, + Qt < t 5 t,
Thus, we have three variables to optimize, SO,t 2 , and s*, subject to the restriction of fxed substrate feed: -0
2
6
4
8
10
time, h
Figure 3. Dynamics of the optimal process for the simplified model using a two-stage (a) or three-stage (b) control strategy. [Notation: - - -, biomass; -, product; -, inducer; - -, control variable. Parameters: xo = 0.7,umar= 0.5, k~ = 5, qmax = 0.36, ki = 2.98, tf = 10; for three-stage solution, tl = 7.5 and t z = 7.8, and for the two-stage solution, t z = 6.6.1 14
r I
"0
1
I
feed pulse
2
4
6
X
10
tiiiiz, h
Figure 4. Dynamics of fed-batch cultivation with an improper control strategy (notation the same as for Figure 2).
can be stated as maximizing process productivity at the end of the fermentation,
The problem (eqs 17, 19, and 20)was solved numerically, by a quasi-Newton algorithm. Two optimal trajectories have been considered: a two-stage policy (Figure 5a, optimal solution is SO = 0.46 SUA, t 2 = 5.9 h, and s* = 0.1 SUA,yielding J = 4.42) and a three-stage policy (Figure 5b with SO = 0.47 SUA, tz = 7.98 h, and s* = 0.29 SUA,yielding J = 5.1). The three-stage optimal policy has been employed in a open-loop control algorithm, with substrate supply by feed pulses. The substrate feed of concentration sf = 7 SUA was added in pulses of uf = 5 mL during the appropriate stage. The computed quasi-optimal trajectory At) is shown in Figure 6, along with the experimental results. The maximal luminescence level obtained is 30%higher than that in a batch reactor. This level is still lower than the expected value (Figure 5b): due to technological problems, feeding was started (at 6.4 h) before the predefined time ( t 2 = 7.98 h), resulting in a lower luminescence level. The computer simulation of this run revealed good correspondence between model predictions and experimental results (Figure 6).
Conclusions In this paper, luminous E. coli has been used to develop a simple, unstructured model of an autoinductive process. The model then was used to determine the optimal control for fed-batch operation of a bioreactor and for maximizing metabolite production. The class of functions
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40,
strated experimentally by the comparison of optimal and improper control strategies. Although a luminous strain of E. coli has been used as a simple model of a n autoinductive process, induction and autoinduction of metabolite production are features of many cultures obtained by recombinant DNA technology (e.g., Togna et al., 1993). Moreover, recent studies revealed the distribution of the V. fischeri inducer across a wide range of Gram-positive and Gram-negative genera. In many cultures, the rate of biosynthesis is dependent on the density of the bacterial culture, implying a n autoinductive regulatory control similar to that described in this study (Ulitzw, 1994). Thus, we expect the three-stage control strategy to be applicable for a wide range of technological processes. Such control requires an appropriate model of the system, which can be applied either in a n open-loop mode or in a closed-loop mode if continuous probing of fermentation is available. The outcome of such an operation, however, may be very sensitive to kinetic parameters, which may change from one batch to another. In a future publication, we will show that a feedback-linearizing controller based on a nonlinear state estimator can be employed to maintain the optimal trajectory.
lime, h 40
Notation feed rate (MI)
YI
Hamiltonian inducer concentration +glL) performance index constant (ki = qmJpmar) kinetic constants
$
Y K
0
2
4
6
8
time, h
Figure 5. Dynamics of fed-batch cultivation with the quasioptimal control strategy using a two-stage (a) or three-stage (b) policy. r i , ri,max
P
F
tf U
v
VI
I
a
Vf X
tiine, h
Figure 6. Validation of the quasi-optimal control strategy showing experimental and predicted temporal patterns (notation the same as for Figure 2).
to which the control variable fit) belongs was determined analytically from the analysis of a simplified problem, and the general problem was then solved numerically. The importance of proper process control was demon-
luminescence (LUA) number of fermentations having different initial conditions maintenance coefficient (SU/(gh)) number of measurements in the ith fermentation product activity (LUL) actual and maximal specific product formation rates, respectively (LU/(gh)) actual and maximal specific inducer formation rates, respectively (CLg4g-h)) specific substrate consumption rate (SU/(gh)) substrate concentration (SUA) optimal substrate concentration (SUA) concentration of feed (SUA) time (h) onset of the induction and product synthesis phases, respectively (h) final time (h) control variable volume (L) volume of one feed pulse (mL) cell density CglL) yield coefficients (g/SU, LU/SU, pg/SU, respectively)
Greek Letters coefficients (a = 1, y = 1) a,Y 6t time interval to attain the optimal substrate concentration (h) y1, y2 weight coefficients actual and maximal specific growth rates respecp, ptively (lh) E coefficient in eq 2b (CO)
Prog., 1995, Vol. 11, No. 1 intracellular product activity (LU/g) adjoint variables switching function Subscripts and Superscripts 0 att=O max maximum min minimum e experimental C calculated sing singular X biomass P product i inducer
Acknowledgment The experiments were supported by the Otto Meyerhof Biotechnological Laboratories. B.T. was supported in p a r t by the Scientists’ Absorption Program of the Ministry of Absorption. We thank Dr. Yoval Shoham of our Department of Food Engineering and Biotechnology for fruitful discussions.
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87 Microcoloniesof Luminous E . coli. Biotechnol. Bioeng. 1994, submitted for publication. Hastings, J. W.; Weber, G. Total Quantum Flux of Isotropic Sources. J. Opt. SOC.Am. 1963,53,1410-1415. Lee, S. B.; Bailey, J. E. Genetically Structured Models for lac Promoter-Operator Function in the E. coli Chromosome and in Multicopy Plasmids: lac Operator Function. Biotechnol. Bioeng. 1989,26,1372-1382. Modak, J. M.; Lim, H. C.; Tayeb, Y. J. General Characteristics of Optimal Feed Rate Profiles for Various Fed-Batch Fermentation Processes. Biotechnol. Bioeng. 1986,28, 13961407. Mosrati, R.; Nancib, N.; Boudrant, J. Variation and Modeling of the Probability of Plasmid Loss as a Function of Growth Rate of Plasmid-Bearing Cells of E. coli During Continuous Cultures. Biotechnol. Bioeng. 1993,41,395-404. San, K. J.; Stephanopoulos, G. Optimization of Fed-Batch Penicillin Fermentation: a Case of Singular Optimal Control with State Constraints. Biotechnol. Bioeng. 1989,34, 7278. Sheintuch, M.; Vashitz, 0.;Wolffberg, A. Engineering Applications of Bioluminescence: Modelling of Mass Transfer in a Hollow Fiber and in a Chemostat. Chem. Eng. Sci. 1992,47, 2615-2620. Togna, A. P.; Fu, J.; Shuler, M. L. Use of a Simple Mathematical Model to Predict the Behavior of E. coli Overproducing B-Lactamase within Continuous Single- and Two-Stage Reactor Systems. Biotechnol. Bioeng. 1993,42,557-570. Ulitzur, S.The Regulatory Control of the Bacterial Luminescence System-A New View. J. Biolumin. Chemilumin. 1989, 3,317-325. Ulitzur, S. The role of GroESL and LexA Proteins in the Regulatory Control of the V. fischeri lux System. 1994, submitted for publication. Ulitzur, S.; Kuhn, J. The Transcription of Bacterial Luminescence is Regulated by Sigma 32.J . Biolumin. Chemilumin. 1988,2,81-93. Vashitz, 0.;Ulitzur, S.; Sheintuch, M. Mass Transfer Batch and Continuous Kinetics in a Luminous Strain of X. campestris. Chem. Eng. Sci. 1988,43,1883-1890. Whitney, G. K.;Glick, B. R.; Robinson, C. W. Induction of T4 DNA Ligase in a Recombinant Strain of E . coli. Biotechnol. Bioeng. 1989,33,991-998. Accepted September 14, 1994.@ @Abstractpublished in Advance ACS Abstracts, October 15, 1994.
