A Cybernetic View of Microbial Growth: Modeling of Cells as Optimal Strategists P. Dhurjati, D. Ramkrishna, M. C. Flickinger, and G. T. Tsao School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907 Accepted for Publication February 27, 1984 A cybernetic framework is presented which views microbial response in multiple substrate environments as a judicious investment of cellular resources in synthesizing different key proteins according to an optimal regulatory strategy. A mathematical model is developed within the cybernetic framework for the diauxic growth of Klebsiella pneurnoniae on a mixture of D-glucoseand D-xylose. The “bang-bang” optimal policy describes well the experimental observations obtained using a fermenter coupled to an Apple I1 microcomputer. Striking variations in respiratory levels are observed experimentally during the switching of the cell’s adaptive machinery for the utilization of the less preferred substrate. INTRODUCTION The present work is concerned with modeling the growth of microorganisms in multiple substrate environments. The problem of cell growth in multiple substrate media is of importance in the fermentation of hemicellulose hydrolysates and in such processes as waste water treatment. In such multisubstrate media, the cells exhibit a varied set of.metabolic activities, the most striking of which is manifested in the phenomena of diauxic growth. Here, the organism displays a marked preference among two functionally equivalent substrates by deferring utilization of one of the substrates until complete exhaustion of the other. Such calculated microbial responses to their surroundings provide a considerable challenge to the “chemical-kinetic framework” and it is doubtful whether a proliferation of the number of lumped biochemical species, per se, could account for the broad spectrum of possible microbial responses. In our approach, referred to as the cybernetic [the word “cybernetic” is derived from the Greek “ ~ v & p n p ” which means steersman] approach, we adopt a viewpoint at variance from modeling based purely on kinetic considerations. We consider the cells to be optimal control systems involved in the maximization of a performance index (goal). The main advantage of such an approach is that we have a relatively simpler description of the microorganism and the complex regulatory processes are reflected in terms of the cell’s accomplishment of its optimal control objectives. Having inferred some performance criteria for Biotechnology and Bioengineering, Vol. XXVII, Pp. 1-9 (1985) 01985 John Wiley & Sons, Inc. the cells based on our biological knowledge, we then explore the extent to which observed regulatory phenomena can be modeled without having to detail all the rudimentary physicochemical interactions of the numerous intracellular constituents. Thus, what is being done is the interpretation of an elaborate facility acquired through evolution in terms of what an optimal system would accomplish. Theviewpoint of treating a living entity (such as a bacterium) as an optimal steering system is not alien to biology. Bremerman2 states that “Most biological systems are goal oriented. . . A goal seeking system is called cybernetic if it pursues its goal while interacting with an environment.” A mathematical model is developed within the cybernetic framework for the specific case of the diauxic growth of Klebsiellapneumoniae. The model is solved using Pontryagin’s Maximum Principle and interpreted in the light of experimental results. CYBERNETIC FRAMEWORK The cybernetic view of the cell presumes the existence of certain characteristic features in the “model” microorganism. Thus, we consider the cell to possess an “adaptive machinery” where specific metabolic changes occur in response to variations in the environment and also to have a “permanent machinery” that does not participate “directly” in the adaptive response. The “regulator” which is the third component of the model cell, is an idealization that embodies the control information in the DNA and guides the nature of the adaptive response. A schematic of the cell as a cybernetic system is shown in Figure 1. A detailed description of the various components of the model cell follows. Adaptive Machinery The response of microorganisms to variations in a multiple substrate environment is very specific and usually limited to the biosynthetic pathways of a few key proteins necessary for incorporating the substrate into the cell’s metabolism. The “adaptive machinery” is assumed to be involved in the synthesis of these key proteins and includes CCC 0006-3592/85/010001-09$04.00 r -- ENVIRONMENT SUBSTRATES Figure 1. Schematic of cell as a cybernetic system. a resource facility consisting of the precursor biomolecules required for the key proteins. From a modeling viewpoint, we may consider only a single critical resource such as a particular amino acid to be the resource required for the reaction networks concerned with the biosynthesis of the key proteins. The heart of the cybernetic approach proposed herein lies in the “optimal” allocation of this resource among competing reaction networks for the synthesis of key proteins for the various substrates (Fig. 1). Of course, the most compelling reason for an optimal allocation of the resource is their limited availability. Since metabolic processes must occur in time, we express the constraint on the availability of resources by requiring that the “rate of availability” of the critical resource be no greater than a stipulated maximum (k,,,,,). physiological realization of the regulator within the cell would most likely be the control regions of the operon such as the promoter and operator sites. Apter3 presents an interesting discussion on how the operon could be considered to be a computing system or an automaton. The “program” for utilizing the substrates has associated with it two important features: (1) an Optimization criteria or a guiding strategy for the optimal allocation of resources, and (2) the period for which such an optimal strategy is to be established. There are other examples in the biological literature where optimization criteria have been used in the modeling of biological system^^.^ and detailed discussion on the appropriateness of such assumptions are presented in refs. 3 and 6-8. From the viewpoint of the theory of evolution, an objectivefunction that suggests itself is the maximization of cell mass productivity during a certain “period” of optimization. The term “period” here refers to the time interval for which the optimization criteria are to be established. The implementation of control decisions such as synthesis of proteins involves thousands of reactions and long periods of time (up to a couple of generations) and, consequently, one could assume that such control decisions have associated with it a long-term projection of the future. Thus, one could argue that the metabolic inertia inherent in a microbial system calls upon an efficient organism to prepare “ahead.” The long-term policy is assumed to be initiated as well as reevaluated when “external” changes occur in the concentrations of substrates in the environment that are brought about by other than the growth of the cells themselves. From a biological viewpoint, the evolutionary history (of environmental variations) embodied in the genotype of the specific organism would dictate the nature of its control strategy. In the present endeavor, we wish to “infer” the appropriate combination of optimization and constraints based on our biochemical knowledge so as to best describe (for engineering purposes) the regulatory behavior of the microorganism. Permanent Machinery Once the adaptive machinery responds to the input of environmental conditions with an output of certain key proteins, the permanent machinery carries out the thousands of reactions necessary for the replication of cellular material. The permanent machinery thus consists of metabolic reactions which always occur and it is in this sense that they are “permanent” features of the cell. The nature of reaction networks that are included in the permanent machinery is largely dependent on the regulatory problem under consideration just as the adaptive machinery is determined by the nature of the adaptation. Regulator The regulator is the “decision-making” apparatus of the model cell that senses the environment and decides on a “program” of its utilization by optimally allocating resources to the key protein synthesis systems. The possible 2 THE MATHEMATICAL MODEL The various state differential equations for the mathematical model presented here have been formulated within the context of the cybernetic framework discussed in the previous section. Consider the batch aerobic growth of a microorganism in a multiple substrate environment where the carbon source is the only growth-limiting nutrient and is provided through “n” different carbon substrates. We denote by Si the concentration of the ith-limiting carbon sources expressed as grams per liter of fermenter liquid volume. The concentration of the microorganisms in the fermenter, X , , refers to the dry mass of cells in grams per liter. The physiological state of the cells, represented in terms of the various key proteinsp;, is the concentration of the ith key protein involved in the incorporation of the substrate Si. We can then define an environmental vector given by BIOTECHNOLOGY AND BIOENGINEERING, VOL. 27, JANUARY 1985 Parameter VSiis equivalent to (pmaxi/Ysj) where pmax; is the maximum specific growth rate on the ith substrate. The production of cell mass is the cumulative effect of the utilization of the various substrates, and is given by and a physiological vector given by p = [PI, pz, . . .> P,ll The state vector X provides information regarding the fermenter in terms of the state of the environment and the physiological state of the cells: X = [Xc,S1, * - . , S ~ , P ~ , ***,Pnl PZ~ We represent by Ri the allocation rate of the critical resource to the ith key protein. The fractional allocation given byRj/Rmaxare then denoted as ui and we can define an allocation vector given by u = [UI,u2, * . ., u,l i=l The maximum possible key protein concentration of the ith protein Pi during exponential growth on S; alone is represented by Pi,,, . The nondimensionalized key protein concentration for the ith key protein, denoted by ej is defined by ei = (P,./Pjm,,). The rate of synthesis of ei is considered to be dependent on the availability of resources, uikm,, , in the following manner: dt i=l Y.; ~ dt where Ysj is considered to be the yield of cell mass per unit of substrate Sj for the case where cells are grown on S; alone. As discussed earlier in the section on the cybernetic framework, the regulator has associated with it an optimization criterion. One possible optimization period could be the maximization of cell mass productivity given by i- tr max J O ~ r 0d t where tf is the optimization period defined by the following stopping condition: cuj=l -dei _ -c dt In the present case, we consider R,,, to be a constant. Mandelstam9 has shown that the turnover of protein could indeed lead to a constant pool of amino acids even when no external substrates are consumed. Thus, the constraint on the total availability of resources can be expressed as n dS; n dXC -- - a j f ( u j k m a x) biei In the above, a j is a constant dictating the rate of synthesis, f gives the filnctioiial dependence of the synthesis rate on the fractional allocations u j ,and the last term represents the degradation of proteins (or protein turnover) with bi being a degradation constant. For our present problem, we assume a linear functional dependence and include R,,, in the constant ai. Once the adaptive machinery synthesizes the appropriate key proteins, the permanent reactions for the growth proceed in an invariant manner and can be represented as si + Pj + x, x, + The above is a macroscopic representation of the cell growth process and does not reflect the actual stoichiometry. The rate of incorporation of substrate S; by key protein Pi can be described by simple Monod-type kinetics: 1 dSj - VSieiSi X , dt kSi Sj + Here, Vsi and k,,are growth and saturation constants, respectively, of Monod type and the negative sign is present because substrate Si is being consumed by the cells X , . n C Sio i= 1 + se S;dt = E 0 where Sjois the amount of substrate Sj at time t = 0. OPTIMAL SOLUTION FOR A SPECIFIC CASE The solution of the optimization problem discussed in the previous section requires the use of calculus of variation. Specifically, we solve the problem by using Pontryagin’s maximum principlelO,llwhich involves the definition of a Hamiltonian and a set of adjoint differential equations corresponding to the state variables. The numerical techniques for the solution of these optimization problems are discussed by Denn’’ and Bryson and Ho.I2 One first guesses an initial optimal u, solves the state equation forward in time, and then utilizes the state solution to solve the adjoint differential backward in time. The value of u is then updated using the steep descent method until no further improvement in the performance index is obtained. The values of the parameters for the specific case of Klebsiellapneurnoniae growing aerobically in a batch culture consisting of two limiting carbon sources (D-glucose and D-xylose) are summarized in Table I. The initial glucose and xylose concentrations were 1 and 2 g/L, respectively. The values of growth rates and yields were those obtained from single substrate experiments while values for other parameters such as a; and bi in the key protein equation were assumed. For this problem, the Hamiltonian is linear in u1 and can be represented as H=uq+y DHURJATI ET AL.: CYBERNETIC VIEW OF MICROBIAL GROWTH 3 where a is referred to as the switching function, and depending on its value, we have the following optimal solution u1 = 0 if u > 0 =lifu<O When a = 0, we have a singular solution and the optimal u1 can be obtained by setting the successive derivatives of the switching function, with respect to time, equal to zero. For the present problem, we find that we need not be concerned with a singular solution and the optimal solution turns out to be what is referred to as a “bang-bang’’ optimal policy. Thus, as shown in Figure 2, the optimal u , Table I. Table of values. Parameters Pmaxl Pmax2 YS! ys2 a yo2 a Go, yco, ks, ks, 1 Values Parameters 1.07 h-’ 0.51 h-’ 0.525 0.36 1.754 1.10 1.35 0.89 10 mg/L SI(0) Sz(0) el@) ez(O) 01 7 b, a 2 . b, x,(0) kLac* V aD-glUCOSe. bD-xylose. Values 1.0 g/L 2.0 g/L 1 .o 0.002 10 h-’ 1.0h-’ 0.0175 g/L 0.023 g mol/L h 1.5 L takes on a value equal to 1 followed by a switch in policy to a value u , = 0 at switching time t,, . The optimal policy can be substituted in the state differential equations to yield the solution for the various states as a function of time. Log (cell mass) and substrate concentrations are plotted as function of time in Figure 3. We can see that growth takes place in two distinct stages. The first stage corresponds to the optimal policy u 1 = 1 and consists in the exclusive utilization of glucose with exponential growth of cells. This is followed by a switch in policy and the consumption of xylose. At the point where a discontinuous change exists in the control variable, the WierstrassErdmann corner conditions need to be satisfied, as discussed by Citron.I3 The optimal strategy discussed previously could be detected by experimentally monitoring uptake of oxygen by the cells. In this connection, it is useful to derive relationships between the dissolved oxygen tension (DOT) and the adaptive machinery of the cell represented in terms of the key proteins. This would enable us to utilize DOT to detect alterations in the adaptive machinery of the cell caused by policy changes. An oxygen balance in the culture gives dc/dt = kla(c* - c) - rx where r, is the instantaneous oxygen consumption rate of the cells, kLa is the mass transfer coefficient, c is the concentration of the oxygen in the liquid phase, and c* is the CELL MASS AND STRATEGY VS TIME 1.m ----.l4bDEL -EXP ‘“1 .€m2- L7 c w E .mLo n a -i.m- 5 m Figure2. Optimal strategy and state solution for cell mass comparison with experimental values for Klebsiellapneumoniae growing on 1 g/L D-glUCOSe and 2 g/L D-XyIOSe at 37°C. 4 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 27, JANUARY 1985 LCIG[CELL MASS1 flND SUBSTRATES Figw3. TIME Plot of substrate concentrations and natural logarithm of cell mass versus time. saturation oxygen in the liquid phase. DOT expressed as percent of air saturation is given by lOO(c/c*). The oxygen consumption rate r, may in general be n r, = VS C rxi i= 1 where the index i refers to the ith substrate and rXiis given by 5dSi)ei -XC Yo,(i) where YO2(!) is the yield of cells from oxygen for the ith substrate (in g/g) and is assumed to be a constant. One can adopt a pseudo-steady-state assumption (dc/ dt = 0) in the oxygen balance equation above and obtain an expression for the dissolved oxygen tension as follows: At the instant corresponding to the change in strategy, one could expect the cell massXc and the value of the mass transfer parameter kLac* to be constant. Thus, any changes in DOT would be reflective of changes within the cell which affect the intrinsic growth related parameters or the physiological vector represented in terms of the nondimensionalized key protein level e i . Thus, a switch in cellular strategy should be more readily observed in the dissolved oxygen response in contrast to cell mass which cannot be expected to provide instantaneous indications of the physiological changes. As discussed in the next section, the dissolved oxygen response is indeed a very sensitive experimental tool to diagnose switches in cellular policy. RESULTS AND DISCUSSION The experimental materials and methods are discussed in the Appendix. The experimental plot of cell mass (g/L) as afunction of time is shown in Figure 2. The initial levels of D-glucose and D-xylose in this experiment were 1 and 2 g/L, respectively. As can be seen from this plot, growth takes place in two distinct exponential phases and we have the growth pattern referred to by Monod14 as "diauxie." Also superposed on this plot is the state solution for cell mass resulting from the bang-bang optimal policy. It can be seen that the bang-bang optimal strategy can describe fairly well the diauxic growth of Klebsiellapneumoniae on a glucose-xylose mixture. From the point of the optimal formulation, the cells first allocate resources to the key proteins involved in the exclusive utilization of glucose. This is followed by a switch in policy and an allocation of cellular resources to key proteins for xylose, thus enabling the utilization of the less preferred sugar. While the cell mass measurement does indicate the policy switch, it is not a satisfactory measurement from the cybernetic perspective. This is because corresponding to the change in optimal strategy, the cell mass level does not give us a sensitive indication of the nature of strategy involved. In DHURJATI ET AL.: CYBERNETIC VIEW OF MICROBIAL GROWTH 5 this connection, the dissolved oxygen level as measured by a galvanic oxygen probe turns out to be a sensitive monitor of the cellular strategy. Figure 4 consists of a plot of dissolved oxygen (as percent of air saturation) versus time (in h). The model values were obtained by using the bang-bang strategy and solving for the differential equation describing the change in oxygen level in the fermenter. The first stage of the dissolved oxygen up to just over three hours is quite similar to the DOT curve for the single substrate. But corresponding to the “bang-bang’’ switch in policy, there is a dramatic and sudden variation in the dissolved oxygen which changes from close to 60% to almost saturation levels. According to the model, the switch corresponds to allocation of resources to the key proteins for the less preferred substrate which results in the growth of cells and a consequent decrease in dissolved oxygen level. The dissolved oxygen level drops again until complete exhaustion of the less preferred xylose. At this time, both sugars have been exhausted and the dissolved oxygen levels out at near saturation levels. As can be seen from the plot, the experimentally dissolved oxygen level does not “rise back” as fast as predicted. This is probably due to the production of certain intermediates which are slowly consumed at the end. In such a case, this phenomena could actually be a triple substrate phenomena with the third substrate being an intermediate fermentation product which is subsequently consumed. DUT However, the most pertinent conclusion from the DOT profile is that dissolved oxygen can indeed be a very sensitive tool to diagnose cellular strategies. In addition to the cell mass and dissolved oxygen measurements, the exit gases from the fermenter were also analyzed. The infrared analysis of C 0 2and the paramagnetic measurement of O2 can be utilized along with the instantaneously measured inlet air flow rate to perform on-line material balances on the gases flowing through the fermenter as discussed by C o ~ n e y . These ’~ material balances yield the experimental values for the carbon dioxide evolution rate (CER) and the oxygen uptake rate (OUR) as shown in Figures 5 and 6 , respectively. The carbon dioxide evolution takes place in two stages corresponding to the utilization of glucose and xylose with the switch occurring around 3 h. The oxygen uptake rate shows a similar behavior with cells taking up oxygen in two different stages corresponding to each of the policy periods. The model values for OUR were obtained using the expression for r, derived in the previous section. An equation similar to the oxygen uptake rate equation was written for CER replacing Yo2 with Yco2(,). These stoicfiometric coefficients were estimated from single substrate experiments and are listed in Table I. These expressions, along with the bang-bang strategy, are able to describe in general terms the experimental CER and OUR profiles. The preciseness of these fits can of course be improved with a more elaborate mathemati- VS TIME 150.0 __.. PODEL -EXP 125.0 120.0 la5.0 - 90.0 w ( I I t x 75.0 > v) D E 0 60.0 Y5.0 30 .o 15.0 0 .m I 1.m 2.m I I 3.mTlMEY,.ZuRS) 5.m 1 I I 6.m 7.m 8.mO m CER TIME VS ~ _ MODEL _ . -EXP m OUR VS TIME -...FWDEL -EXP I .m 1.m I 2.m I 3 . m TIME~i&RSI s.m r e.mm 7 .t m 8 .I m n n Figure 6. 0,uptake rate versus time for growth on glucose and xylose. DHURJATI ET AL.: CYBERNETIC VIEW OF MICROBIAL GROWTH 7 cal description of the microbial system, but our main focus is on the conceptual aspects of the model. LIMITATIONS AND ALTERNATIVES The present approach to modeling has endeavored to circumvent microscopic details (such as those occurring at the level of the operon16) by postulating the microorganisms to be cybernetic systems capable of optimally steering themselves toward certain control objectives. The cybernetic framework, however, has the flexibility of accounting for significantly more biological detail in its state differential equation constraints than has been considered in the specific mathematic model. Such elaborations on the model would be most fruitful to the extent that actual experimental measurements can be made which relate to the kind of microscopic variables which are included. The measurement of several of these microscopic variables in fermentation processes is in many cases difficult or impossible, and as such, their inclusion in the model may not be warranted at the present stage. The mathematical model considers only a single intracellular resource to be limiting. A more complete model within the cybernetic framework could account for a diversity of precursor biomolecules. The model makes an assumption of a constant pool of resources available at any time. It is more likely that this resource pool would be influenced by the energy content of cells or by the growth rate of cells. There is a possibility of mutations occurring during the course of the fermentation leading to changes in the genotype of microorganisms. One of the present limitations of the model is that it is not possible to account for such mutative changes because of the assumption of a 4 Cart n Amplifier Boards 1 Analog to Digital Converters Apple II Computer Figure 7. 8 Schematic of experimental setup. constant genotype. By including the genotype itself as a variable, it is conceivable that the cybernetic framework can account for such mutative changes. Some other limitations of the model are in regard to the nature of the mathematical relations adopted. It is possible to explore a wide range of other objective functions with different time scales of optimization within the cybernetic framework. Alternate state differential constraints could also be considered. For example, the state equation for key protein production could be postulated to have a nonlinear dependence on the control variable. This leads to a nonlinear dependence of the Hamiltonian in the control variable which could allow for simultaneous utilization of both sugars. A limitation of the present model is that such simultaneous utilization is permitted only in the event of a singular solution because of the postulated linear dependence on the control variable. CONCLUSIONS We have illustrated here that a cybernetic model is capable of describing regulatory phenomena such as those observed in diauxie. The switching “on” and “off” of operons as described by the operon theory16of Jacob and Monod is interpreted in our model to be simple “bangbang” type of optimal strategy for the cellular system. The experimental results for the batch growth of Klebsiella pneumoniae on D-glUCOSe and D-xylose agree well with a cybernetic description of the intracellular control system. This work was supported in part by National Science Foundation Grant No. ENG 78-20964. The experimental portion of this work was carried out at the Laboratory of Renewable Resources Engineering, Purdue University, Lafayette, IN. APPENDIX: DESCRIPTION OF EXPERIMENTAL MATERIALS AND METHODS All experiments were performed with the microorganism Klebsiella pneumoniae ATCC 8724 obtained from the American Type Culture Collection (Rockville, MD). The bacterium was maintained on agar slants with xylose as the sole carbon source. The medium utilized for growing the organism is listed in Table A I . All innocula were prepared in 250-mL Erhlenmeyer flasks with a culture volume of 50 mL at 370°C using glucose as the carbon source. A 2-L New Brunswick fermenter (culture volume = 1.5 L) with a magnetic coupled drive was used in all experiments. The power for agitation was provided by 1/8-hp motor (Model 42140 Dayton Mfg. Co., IL) and the agitation rate was 900 rpm. Dissolved oxygen tension (DOT) was measured using a galvanic oxygen electrode fitted with a 1 -mil Teflon membrane. The probe was constructed according to the specifications of Borkowskii7 and had a response time of less than a minute. In our entire discussion, “cell mass” referred to the dry mass of cells in grams per liter. Cell mass was monitored using optical density measurements at a wavelength of 540 nm by means of a Bausch & Lamb spectrophotometer (Spec 20) adapted with a special flow dilution device. The inlet air flow was measured using an air flow transducer (model 8141, Matheson, NJ). The CO, was measured using an infrared analyzer (model IR-702, Infrared Industries, Inc., CA) while oxygen was measured with a paramagnetic analyzer (model 755, Beckman, NJ). More details of the off-gas analysis equipment are presented by Flickinger.” ‘* BIOTECHNOLOGY AND BIOENGINEERING, VOL. 27, JANUARY 1985 X X, state vector concentration of cells (g/L) stoichiometric coefficient for substrate S, (g cells/g C 0 2 ) yco2c, stoichiometric coefficient for substrate S, (g cells/g 0 2 ) YOZCd growth yield constant for substrate S, (g cells/g S, ) Ys, Table AI. Composition of the medium.a Component K2HPO4 3HzO K2HP04 (NH4 )2. so4 NaCl CaC12.2H20 FeSO,. 7H20 MnS04. 4 H 2 0 ZnS04.7H20 EDTA MgS04 7H2O Concentration (g/L) 11.4 1.5 3.0 0.1 0.014 0.01 0.0028 0.0075 0.4 0.24 Greek switching function specific growth rate as a function of S, pmaxr maximum specific growth rate for S, (h-' ) Ll p,(S,) Superscripts and subscripts differentiation with respect to time 1 refers to the ith substrate aExcludingthe carbon source, it is the same as the medium of Anderson.22 The recalibration of the off-gas equipment as well as continuous monitoring of all data was done by means of an Apple I1 Plus computer as discussed by Forrest2' and DhurjatL2' NOMENCLATURE constant for key protein i synthesis rate (h-') constant for key protein i breakdown rate (h-l) concentration of dissolved O2 in liquid (g/L) saturation concentration of dissolved 0,in liquid (g/L) dissolved oxygen tension (7' of air saturation) fraction of maximum possible key protein P, Hamiltonian oxygen solution rate constant (h-') saturation constant for substrate S, (g/L) number of limiting substrates structural vector of key protein levels key protein for substrates S, (g/g biomass) maximum level of key protein P, rate of allocation of critical resource R for synthesis of key protein P, maximum rate of availability of critical resource cumulative oxygen uptake rate of the culture (g/L h) oxygen uptake rate due to utilization of substrate S, environmental vector of limiting substrate concentrations concentration of limiting substrate i (g/L) initial concentration of limiting substrate i (g/L) time (h) time period of optimization (h) switching time (h) decision vector fraction allocation of resources to P, maximum uptake rate of substrate S, (h-l) References 1. 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