Unveiling Steady-State Multiplicity in Hybridoma Cultures: The Cybernetic Approach

Transcription

1 Unveiling Steady-State Multipliity in Hybridoma Cultures: The Cyberneti Approah Abhijit Anand Namjoshi, 1 Wei-Shou Hu, 2 Doraiswami Ramkrishna Chemial Engineering Building, Purdue University, West Lafayette, Indiana 47907; telephone: ; fax: ; ramkrish@en.purdue.edu 2 Chemial Engineering and Materials Siene Department, University of Minnesota, Minneapolis, Minnesota Reeived 5 April 2002; aepted 5 June 2002 DOI: /bit Abstrat: Mammalian ells grown in suspension produe waste metabolites suh as latate, alanine, and ammonia, whih redue the yield of ell mass and the desired produt on the nutrients supplied. Previous studies (Cruz et al., 1999; Europa et al., 2000; Follstad et al., 1999) have shown that the ells an be made to alter their metabolism by starving them on their nutrients in ontinuous ultures at low dilution rates or starting the ulture as a fed-bath. This leads to multiple steady states in ontinuous reators, with some states being more favorable than others. Mathematial models that take into aount the metaboli regulation that leads to these multiple steady states are invaluable tools for bioreator ontrol. In this artile we present a yberneti modeling strategy in whih Metaboli Flux Analysis (MFA) is used to guide the yberneti formulation. The hybridoma model presented as a result of this strategy onsiders the partially substitutable, partially omplementary nature of gluose and glutamine. The hoie of ompetitions within the network is guided by MFA and the model is suessful in explaining the three multiple steady states observed. The yberneti model though identified for the hybridoma experiments of Hu and others (Europa et al., 2000) seem generally appliable to mammalian systems as it aptures the pathways that are ommon to mammalian ells grown in suspension. The model presented here ould be used for start-up strategies for ontinuous reators and model-based feedbak ontrol for maintaining high produtivity of the reator Wiley Periodials, In. Biotehnol Bioeng 81: 80 91, Keywords: Mammalian ells; metaboli flux analysis, yberneti modeling strategy; hybridoma ultures; multipliity INTRODUCTION Correspondene to: Doraiswami Ramkrishna Contrat grant sponsor: NSF Contrat grant number: CTS Hybridoma ells when ultured on a well-defined medium omprising gluose, glutamine and other amino aids produe ell mass, monolonal antibodies and waste metabolites suh as latate, alanine and ammonia (Hu and Oberg, 1990). Hybridoma ells [in partiular, and mammalian ells in general (Cruz et al., 1999)] display multiple steady states with widely varying onentrations of ell mass, desired produt as well as waste metabolites (Follstad et al., 1999; Zhou et al., 1997). This means that for idential input onditions to a ontinuous reator, the outlet onditions hange depending on how the ulture is made ontinuous. These multiple states are manifestations of the omplex interation between ells and their environment. Cells hannel substrates through myriads of intraellular reations to generate new ell mass and energy and in the proess generate and exrete different byproduts. These ellular reations, termed metaboli pathways are the key to understanding ellular behavior. What makes this proess diffiult is the additional level of omplexity present in biologial systems in ontrast with purely hemial systems beause of the geneti information present in living ells. The geneti ode not only alters the extents of different reations but also the reations themselves, by induing (and ativating) new enzymes while repressing (and deativating) existing ones, in response to hanges in the abioti environment. This underlying phenomenon of ellular regulation ditates the nonlinear behavior in biologial systems and is the hief reason of multipliity in ontinuous bioreators. Here we address the peuliar features of hybridoma ell growth in bath, fed-bath, and ontinuous operating onditions and onstrut a mathematial model apable of explaining the observed phenomenon. The mathematial models available in literature (Brian et al., 1989; Bree et al., 1988; Glaken et al., 1988; Guardia et al., 2000; Miller et al., 1988), with one exeption, do not address this phenomena of multiple steady states. The yberneti modeling approah (Kompala et al., 1986; Ramkrishna, 1982; Ramkrishna et al., 1987) is ommissioned toward this task beause yberneti models have, over the years, been extremely suessful in modeling ellular regulation (Baloo and Ramkrishna, 1991a; 1991b; Jones and Kompala, 1999). We also illustrate how model 2002 Wiley Periodials, In.

2 onstrution is failitated by a deeper understanding of the distribution of metaboli fluxes at different steady states. Thus, unlike previous yberneti efforts, this work signals a shift in paradigm towards the deployment of Metaboli Flux Analysis (MFA) for the onstrution of yberneti models. Suh a dynami model should serve as an invaluable tool towards understanding ellular metabolism and devising ell ulture proesses. MULTIPLICITY IN HYBRIDOMA REACTORS When ultured in a bath reator with high initial onentrations of substrates, hybridoma ells produed large onentrations of latate and other waste produts. Suh a reator when shifted to a ontinuous mode, led to a steady state of omparatively low ell mass, and high latate as denoted (diamonds) in Figures The same ells, in another experiment were ultured as a bath with muh lower onentrations of gluose and glutamine and then shifted to a fed-bath in whih the onentrations of gluose and glutamine were maintained at low levels. This ulture when shifted to a ontinuous regime led to a steady state of high ell onentration and low onentration of waste metabolites (shown by empty irles). An intermediate steady state (asterisks) was also reported. The steady state of high ell mass produed more antibodies and denotes an effiient use of substrates indiated also by lower prodution of waste metabolites; this is the desired operating ondition. However, the presene of less-desired steady states at the same inlet onditions makes this bioreator a hallenging problem for ontrol. Reator startup and steady-state operation an be largely failitated by the development of a mathematial model. Metaboli flux analysis is a tool that an help identify rearrangements of fluxes inside the ell at different steady states and therefore an be used to identify a yberneti hybridoma model, as detailed below. ABSTRACTION OF METABOLIC PATHWAYS Metaboli Pathways in Hybridoma Cells The hief metaboli pathways in hybridoma ells involving the primary substrates gluose and glutamine are as follows: Glyolysis and Pyruvate Degradation Glyolysis is the degradation of gluose to yield pyruvate in a series of reations that onsume the oenzyme niotinamide adenine nuleotide (NAD ) and release the redued form NADH and adenosine tri phosphate (ATP). It is the dominant energy yielding metabolism under anaerobi onditions: Gluose + 2 ADP + 2 NAD + 2 P i 2 Pyruvate + 2 ATP + 2 NADH + 2 H + 2 H 2 O (1) Pyruvate an supply aetyl oenzyme A (ACoA) that enters the tri-arboxyli aid (TCA) yle under anaerobi onditions. Pyruvate may also be degraded anaerobially to latate: 2 Pyruvate +2NADH +2H 2 Latate +2NAD (2) For glyolysis to operate ontinuously, the ell must be able to regenerate NAD. Otherwise, all the oenzyme (NAD) will rapidly aumulate in the redued form (NADH) and glyolysis will stop. Under aerobi onditions, most of the NADH is oxidized to NAD by oxidative phosphorylation, whih requires moleular oxygen and takes plae inside the mitohondria. Under anaerobi onditions however, the synthesis of latate is the dominant method of NAD regeneration. Under high gluose onentrations, oxidative phosphorylation an be strethed to the fullest and the redution of pyruvate to latate must also our. In mammalian bodies, the latate formed an be reonverted to gluose by gluoneogenesis, whih involves the atabolism of fatty, aids in the liver [Cori yle (Horton et al., 1996)], but in suspended ells, latate is simply exreted. Pentose Phosphate Pathway The pentose phosphate pathway has two primary funtions: the prodution of NADPH (whih is required for growth) and ribose 5-phosphate. NADPH is the pyridine nuleotide used for redutive biosynthesis. Ribose 5-phosphate is required for the biosynthesis of ribonuleotides and their derivatives, whih are inorporated into ribonulei aids (RNA), deoxyribonulei aids (DNA), and ertain oenzymes. Net reation: Gluose 6 phosphate + 2 NADP + H 2 O Ribose 5 phosphate + 2 NADPH + CO H (3) The Tri Carboxyli Aid (TCA) Cyle The TCA yle oxidizes the aetyl arbon atoms in ACoA ompletely to CO 2 and stores the energy released by onverting the NAD to NADH as well as produing ATP in the proess. The arbon atoms from gluose an thus be utilized for energy prodution via ACoA. In hybridoma ells however, glutamine is the major soure of energy (Batt and Kompala, 1989), espeially under gluose starvation. Glutamine an release an ammonia moleule to form glutamate moiety whih an then form -ketoglutarate and enter the TCA yle. The TCA yle intermediates also supply some of the arbon-based building bloks suh as fatty aids and steroids. Amino aids other than glutamine also interat with the TCA-yle intermediates. Protein Formation Amino aids, inluding glutamine, an be inorporated into biomass as proteins, failitating nitrogen inorporation into biomass. In summary, therefore, gluose supplies the arbon atoms (pyruvate as well as nulei aids: RNA and DNA) neessary for ell growth and an also at as an energy soure. Glutamine along with other essential and nonessential NAMJOSHI ET AL.: STEADY-STATE MULTIPLICITY IN HYBRIDOMA CULTURES 81

3 amino aids is inorporated into the ell as a basi building blok for ellular protein. In hybridoma ells, glutamine apart from being the major soure of energy also ontributes to antibody prodution (Batt and Kompala, 1989). Gluose and glutamine are thus substitutable substrates with respet to energy supply but omplementary due to their unique ontributions to biomass. Gluose and glutamine are thus partially substitutable, partially omplementary substrates. Metaboli Flux Analysis The results of the Metaboli Flux Analysis (MFA), performed by Europa et al. (2000; A. Gambhir, personal ommuniation) are presented in Figure 1. The three histograms from left to right, for every step, show fluxes [in terms of moles/(gdw hour)] in low, intermediate, and high biomass steady states, respetively. In their analysis, a fixed stoihiometri relationship is assumed between biomass formation and substrate onsumption. Hene, the fluxes represented in the other pathways suh as glyolysis and TCA are not true fluxes but the exess values over those required for biomass formation. Note that the flux in latate formation is proportional to that in glyolysis. Also signifiant are the differential fluxes of glutamine into the TCA yle in the three steady states. The fluxes of gluose and glutamine into biomass are nearly onstant, as the biomass formation rate (equal to the dilution rate in a ontinuous ulture) is almost equal ( /h) in all three states. It is also noteworthy that all the fluxes in the TCA yle are more or less onstant in a given steady state, whih means that the ell regulates the yle to prevent aumulation of any TCA yle intermediate. The fluxes are not exatly idential as other amino aids (not shown) enter the TCA yle at different entry points. The results of MFA will be used in the identifiation of the yberneti model, as detailed in the next setion. CYBERNETIC MODEL FORMULATION Cyberneti Framework The yberneti approah (Kompala et al., 1986; Ramkrishna, 1987; Ramkrishna et al., 1987) hypothesizes living ells to be goal seeking and attributes objetive funtions to ellular proesses that share a ommon resoure. Typially, these omprise metaboli units (linear, diverging, onverging, or yli pathways) whih are ellular pathways that share intermediates (Straight and Ramkrishna, 1994). The resoure might be a material resoure (say biomoleules for enzyme synthesis) or temporal resoure suh as transription or translation time. This resoure alloation is refleted in the definition of yberneti u and v variables whih modify the kineti expressions for enzyme synthesis rate and enzyme ativity, respetively. The mathematial expressions for u and v are derived from mathing law and proportional law, respetively (Kompala et al., 1986), by alloating an objetive funtion to a metaboli unit in a pathway based on the pereived (or proposed) ompetition (for resoure) between the elements of the unit. Consider, for instane, a onvergent pathway where intermediates M 1 and S 2 form intermediate M 2. A substitutable ompetition an be hosen to model this proess, where the objetive is to maximize the prodution of M 2. The formulation is summarized in Table I. Cyberneti models an be formulated at various levels of network omplexity by onsidering elementary, loal, and global yberneti variables as disussed by Varner and Ramkrishna (1999a; 1999b). Elementary units are metaboli pathways for whih the resoure ompetitions an be identified and the optimization problem solved to generate the yberneti variables by using the mathing law and the proportional law. Loal units are overlapping elementary units where the yberneti variables are postulated to be the produt of the partiipating pathways for the ommon reations. Global yberneti variables are introdued to aount for the oordinate ontrol of metaboli pathways by sharing of oenzymes (for example, NAD and NADH in latate formation and glyolysis). The stepwise proedure in identifying yberneti models an thus be summarized as: 1. Suitable simplifiation of the metaboli network to apture the phenomena of interest. 2. Identifiation of elementary, loal, and global units in the simplified metaboli network. 3. Identifiation of resoure ompetitions in the metaboli units. 4. Formulation of yberneti variables. 5. Writing speies balanes for the pertinent reator onfiguration. Rational Abstration of Metaboli Networks Flux Viewpoint Figure 1. Metaboli map in hybridoma with fluxes in low, intermediate, and high biomass steady states. Mathematial models involving a limited number of reations are more tratable than ones that apture detailed pathways. A simple model that explains the phenomena being investigated, one identified, an then be expanded to apture detailed flux measurements. This is a top-down approah, in ontrast with a bottom-up approah that seeks to model every observable reation in the ell to explain 82 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 81, NO. 1, JANUARY 5, 2003

4 Table I. Cyberneti formulation for a substitutable ompetition in a onvergent pathway. Reation Unmodified reation rate Unmodified enzyme synthesis Cyberneti variable u Cyberneti variable v Modified reation rate Modified enzyme synthesis 1 2 r 1 = 1m 1 e 1 k 1 + m 1 r e1 = 1m 1 k e1 + m 1 r 1 r 1 + r 2 r 1 max r 1, r 2 r 2 = 2s 2 e 2 k 2 + s 2 r e2 = 2s 2 k e2 + s 2 r 2 r 1 + r 2 r 2 max r 1,r 2 r 1 v 1 r e1 u 1 r 2 v 2 r e2 u 2 experimental results, and then redue it using model redution tehniques to make it omputationally attrative for ontrol purposes. A top-down approah is neessary as the quantifiation of ellular reations that would be required of the bottom-up approah annot be easily aomplished. We now present a flux-based model abstration rationale. As an be seen from Figure 1 gluose (S 1 ) an be hanneled either into glyolysis to form Pyruvate (M 1 ) or into the Pentose Phosphate Pathway, to form intermediates (M 3 ) (Fig. 2). Pyruvate an either enter the TCA yle (represented by intermediates M 2 ) as aetyl oenzyme A and be oxidized into CO 2 to release energy or be metabolized into latate (L), a proess that restores the redox balane between NAD and NADH. Oxaloaetate among other TCA intermediates (M 2 ) an also ontribute to the intraellular pyruvate pool. The seond major substrate, glutamine an deaminate to enter the TCA yle or form the bulk of proteins (M 4 ). The growth preursors M 1, M 2, M 3, M 4 an be onsidered to form the rest of biomass denoted by Ć. This ensures that both substrates S 1 and S 2 are required for the formation of ell mass, sine gluose annot make M 4 and glutamine M 3. Growth is onsidered to be the net aumulation of all intraellular intermediates M 1 to M 4 and Ć. Other amino aids also enter the TCA yle at different points and also reat with pyruvate. Suh interations though important are not modeled presently. Gluose and glutamine are thus partially substitutable in that they an both ontribute towards energy but are partially omplementary towards prodution of riboses and proteins. The model proposed in the urrent work overomes the defiienies in the model of Guardia et al. (2000) (Fig. 3). The model of Guardia et al. (2000) onsiders glyolysis, (S 1 M 1 ) and pentose phosphate pathway (S 1 M 3 ) atalyzed by the key enzymes E 1 and E 3 respetively. M 1 an form the TCA yle intermediates (M 2 ) in a reation atalyzed by key enzyme E 4 or be formed from M 2 using E 5. M 2 represent the nitrogenous preursors as well and an be also formed from glutamine S 2. M 1, M 2 and M 3 together form biomass X in a saturation manner. The network of Guardia et al. (2000) network illustrates pathway abstration but has the following drawbaks: 1. It does not explain all three experimentally observed steady states. 2. The substrate S 1 is the sole soure of M 3, and hene is required for growth, but an supply the other growth preursors M 1 and M 2 as well. As a result, S 2 is not required for the formation of biomass. Thus, the model of Guardia et al. (2000) fails to apture the partially substitutable partially omplementary nature of gluose and glutamine. 3. The formation of latate, whih is a key waste metabolite, is not onsidered expliitly but only as the intraellular aumulation of M It as a tehnial flaw in that the ell-speifi onentrations of intraellular intermediates are not onstrained to Figure 2. The yberneti hybridoma model proposed in the urrent work. Figure 3. Cyberneti hybridoma model by Guardia et al. (2000). NAMJOSHI ET AL.: STEADY-STATE MULTIPLICITY IN HYBRIDOMA CULTURES 83

5 be lower than unity and hene may aumulate to values higher than one. Identifiation of Units, Competitions, and Formulation of Cyberneti Variables The abstrated metaboli network in the hybridoma model an be thought of two subnetworks shown in Figure 4. Network 1 omprises an elementary metaboli unit made up of two enzyme systems E 1a and E 1b that ompete for the utilization of S 1 to produe M 1. Experimental evidene (MFA results) as well as theoretial onsiderations in terms of the ofator requirements (interonversion between NAD and NADH) suggest that the formation of pyruvate from gluose, and latate from pyruvate our onurrently. Hene, the reation M 1 going to L is also onsidered to be a part of this subnetwork. The seond subnetwork is a loal unit that onsists of two elementary pathways. One of them is a divergent pathway leading to the formation of M 2 and M 4 from S 2 while the seond one is a onvergent pathway forming M 2 from either S 2 or M 1. The formulation of metaboli units is thus based on flux observations and hypotheses for the regulation of metaboli pathways. Competitions are then assigned to every elementary unit and the loal and global yberneti variables are formulated as follows. Consider the first subnetwork shown in Figure 4. The first step in glyolysis is atalyzed by about four isoenzymes, with different K m value for the substrate, gluokinase being one with the highest K m value. Experimental observations in the form of fluxes (Fig. 1) indiate that the flux through glyolysis is the highest in the low biomass steady state, and progressively lower in the low biomass steady states. This phenomenon is aptured in the yberneti framework by onsidering two enzymes to atalyze glyolysis, whih ompete in a substitutable manner. Enzyme E 1a is proposed to win the ompetition under onditions of exess gluose, while E 1b takes over under gluose starvation onditions. The yberneti variable u 1a is also used as a global yberneti variable for enzyme E 6 for pyruvate to latate onversion. Thus, under ample gluose onditions, u 1a wins and this leads to latate formation, while under gluose starvation u 1b triumphs, thereby utting down latate synthesis. The pentose phosphate pathway is modeled to be fully saturated with the enzymes and does not ompete ybernetially for gluose utilization. It is modeled largely to predit behavior under gluose-limiting, glutamine-plenty onditions, Figure 4. Subnetworks 1 and 2 in the urrent hybridoma model. as it produes the M 4 moiety whih annot be produed by glutamine. Also, the S 2 M 4 pathway produes proteins whih annot be made from gluose. The presene of the two pathways (S 1 M 3 and S 2 M 4 ) makes the two substrates omplementary. The proesses S 2 M 2 and S 2 M 4 ould be modeled as a omplementary ompetition that maximizes the produt of M 2 and M 4. However, from experimental results (Fig. 1) we see that the flux of glutamine into the TCA yle in the high biomass steady state (denoted by the blak histogram) is negligible with respet to the flux in the biomass (protein) formation step in the same steady state. This indiates that in this metaboli state the protein formation pathway (S 2 M 4 ) has ompletely dominated, or in yberneti terms won over the ompeting (S 2 M 2 ) pathway. The omplementary ompetition ensures that both produts are produed by the pathway and hene does not allow omplete dominane of one pathway over another. The substitutable ompetition promotes a pathway whih has a higher substrate onsumption rate at a given substrate onentration. This means that while one pathway (S 2 M 4 ) may be promoted at low S 2 onentrations, the other pathway (S 2 M 2 ) ould be promoted at higher onentrations. A separate S 2 M 4 flux that is saturated with respet to its enzymes ensures a basal level of M 4 for biomass synthesis. Also, M 1 and S 2 ompete substitutably to produe M 2. M 1 M 4 lead to the formation of the rest of biomass, depited by Ć. Ć, in turn, atalyzes all intraellular reations shown, thereby ensuring that S 1 and S 2 are indeed aptured as partially substitutable partially omplementary nutrients. The speies balanes for all the ases are detailed in the Appendix. PARAMETER ESTIMATION Simulations of the yberneti hybridoma model without substrate inhibition kinetis for the simple bath ase led to striking defiienies in the overpredition of the biomass onentration in the transients as well as the higher values of latate, at steady state. In an attempt to overome the underlying kineti defiieny, a simple unstrutured model was onstruted as shown: ds 1 dt = 1s 1 k 1 + s 1 x 2s 1 k 2 + s 1 x + D s 1 f s 1 (1) dx dt = Y 1 s 1 1 x Dx (2) k 1 + s 1 dl dt = Y 2 s 1 2 x Dl (3) k 2 + s 1 Two simulations of this model for the bath ulture followed by ontinuous ulture are shown in Figure 5. All simulations desribed in this artile were arried out using the stiff ordinary differential equation solvers ode23s or ode15s in Matlab. It appears that the maximum growth rate onstant 1 that fits bath data results in a muh lower (than observed) biomass onentration in the ontinuous ulture. 84 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 81, NO. 1, JANUARY 5, 2003

6 Figure 5. Preditions of an unstrutured model without substrate inhibition kinetis. However, if 1 is inreased to math the ontinuous ulture predition, it results in an overpredition of ell mass in the bath ulture. Thus, it seems that there is some kind of a substrate inhibition in the bath, that is relieved in the ontinuous ulture whih has muh lower substrate onentrations than the bath. To verify this hypothesis, substrate inhibition kinetis of the form s s 1 K 1 + s 1 + k 1 was inorporated and the parameters re-evaluated to give the preditions shown in Figure 6. Hene, substrate inhibition kinetis is used in the urrent hybridoma model. The reation rate onstants are determined similarly by omparing the model simulations with experimental data. The yield oeffiients are alulated from the stoihiometries on the intraellular speies, while the enzyme onstants are alulated from previous yberneti modeling efforts. RESULTS AND DISCUSSION Although our main interest is in steady states of ontinuous reators, insofar as the start-up strategies involve both bath and fed-bath modes, we report alulations for the same in addition to the ontinuous reators. Experimental data are represented as disrete points (with disriminating symbols for different steady states) while model simulations are depited as dotted, dashed, and ontinuous lines in Figures 7 to 11. The intermediate profiles are a onnetion between Figure 7. Gluose onentration in the three steady states (dots denote experimental data). the various reations and are shown in Figure 13. In most modeling of metaboli systems, it is ommon pratie to restrit attention to a hosen set of ellular intermediates and a orresponding restrited set of hemial reations. The apaity of the model to aurately represent the levels of every hosen intermediate is therefore not assured. Some of the intermediates (e.g., pyruvate) are essential to desribe underlying metaboli trends but insofar as all reations in whih they partiipate are not inluded, their levels annot be expeted to remain the same. It is possible to manipulate the levels without affeting the model preditions of those quantities that are measured by either saling rate and saturation onstants or by allowing redistribution of omponents among the rest of biomass. Neither proedure has been adopted here for those omponents that are not measured. The model, with appropriate hoie of parameter values, predits the three experimentally observed steady states as shown. The solid line depits the simulation of a ulture Figure 6. kinetis. Preditions of an unstrutured model with substrate inhibition Figure 8. Glutamine onentration in the three steady states (dots denote experimental data). NAMJOSHI ET AL.: STEADY-STATE MULTIPLICITY IN HYBRIDOMA CULTURES 85

7 Figure 9. Biomass onentration in the three steady states (dots denote experimental data). Figure 11. Cyberneti variables in the three steady states. initiated as a bath with the initial levels of gluose, glutamine and ell mass idential to that in the bath experiment. It is subsequently shifted to the ontinuous mode at 45 h with a dilution rate of 0.031/h. All the enzymes are assumed to be expressed to the maximum extent, exept the enzyme E 1a whih is assumed to be expressed to 95% of the maximum level e max, while enzyme E 1b is assumed to be expressed to 5% of e max. Similarly, e 3 e max and E 7 is assumed to be 20% of e max. This is beause the ells are maintained in a situation of substrate abundane prior to the experiment and hene an be expeted to be in a metaboli state favoring E 1a over E 1b and E 3 over E 7, respetively (see the Appendix for details on initial onditions). A look at the yberneti variables (Fig. 11) provides insight into the mehanism that produes multiple steady states. In a bath ulture with high s 1 and s 2, the steps 1a and 3 dominate 1b and 7, respetively as a result of whih yberneti variables u 1a (solid line, overlayed by the dotted line) and u 3 (solid line) are lose to unity, and as the ulture reahes steady state, remain very lose to 1. On the other hand, variables u 1b and u 7 get dominated by their stronger ounterparts and quikly deay to zero values, and stay there at steady state. The enzyme profiles (Fig. 12) are not very different from the yberneti variables. Thus, enzyme E 1b is quikly degraded in this steady state, while enzyme E 1a is redued in amount ompared to the initial ondition but ontinues to be expressed. Its value is less than e max beause the substrate has not ompletely saturated its high k e1a value. Similarly, E 3 ontinues to be expressed to almost the maximum possible level while E 7 is degraded. Sine enzyme E 3 also ompetes with E 4 and omes out to be a winner, E 4 is also degraded in this steady state. Enzyme E 6 is driven by the same yberneti variable as E 1a and hene, shows a similar profile leading to prodution of latate. The biomass urve shows exellent agreement with the experimental values in the transient as well as steady state, while the latate preditions fall slightly short of the experimental values in the transient. Consider now the fed-bath ulture denoted by the dashed line wherein gluose is ontrolled to low levels. This Figure 10. Latate onentration in the three steady states (dots denote experimental data). Figure 12. Enzyme onentrations g/gdw in the three steady states. 86 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 81, NO. 1, JANUARY 5, 2003

8 Figure 13. Intermediate onentrations g/gdw in the three steady states. ulture is started as a bath with low onentrations of substrates, and then shifted to a fed-bath ulture quikly as soon as the gluose onentration reahes 0.05 g/l. During the fed-bath, gluose onentration is maintained at this level by ontroling the flow rate for about 105 h at whih, the ulture is shifted to a ontinuous mode with a dilution rate of 0.033/h. Notie that in this ulture, u d 7 during the fed-bath, the yberneti variables u 1b and u d 7 dominate their ompetitors u 1a and u d 3, respetively. This is beause both substrates are maintained at low values; this favors reations 1b and 7 over 1a and 3. Although s 1 is the substrate that is ontrolled to low levels, s 2 also gets maintained to low levels, and so this ulture, when shifted to a ontinuous mode results in a steady state of high ell mass and very little latate. Thus, enzyme E 1a is quikly degraded in this steady state, while enzyme E 1b is fully expressed, ontrary to the bath ase just disussed. Also, E 3 loses the ompetition to E 7 and gets degraded. Sine enzyme E 3 also ompetes with E 4, E 4 gets expressed and beomes the dominant soure of M 2. Enzyme E 6 one again behaves similar to E 1a, is down-regulated and hene, there is negligible latate prodution. Thus, the fed-bath is a onditioning mehanism whih helps alter the metaboli state of the ells by enouraging reations that lead to a more effiient utilization of the substrates, as these reations lead to a better yield of intraellular omponents, of ell mass ompared to the reations they dominate. For instane, reation 7 leads to protein formation, while reation 3 leads to deamination, leading to a loss of the nitrogen moiety. Fous now on the simulation denoted by the dotted line. This simulation provides a plausible explanation of the third steady state, for whih the transient data were not available. Suh a steady state ould be simulated by starting the reator as a bath and then swithing to a fed-bath mode before operating ontinuously with a dilution rate of 0.033/h as in simulation 2. The differene is, however, in the onentration of S 2 at the beginning of the bath and the onentration of S 1 maintained in the fed-bath (0.12 g/l). As a result of this, one an see that reation 1a dominates reation 1b, beause of the high level of S 1 that simulates a bath-like senario of nutrient abundane in the fed-bath. However, beause of the low onentration of S 2 in the initial ondition, S 2 remains at a low level in the fed-bath, and so reation 7 ontinues to dominate reation 3. Thus enzyme E 1b is quikly degraded in this steady state, while enzyme E 1a is redued in amount ompared to the initial ondition but ontinues to be expressed, similar to a bath. On the other hand, E 3 loses the ompetition to E 7 and gets degraded and one again E 4 beomes the dominant soure of M 2. Enzyme E 6 one again behaves similar to E 1a, and this leads to latate prodution. This produes a steady state of intermediate levels of biomass as well as latate. Notie that this steady state is shifted to a ontinuous mode muh earlier than the seond simulation (about 65 h), beause if the ell onentration is allowed to build up signifiantly, the shift to a ontinuous ulture an produe a situation of nutrient starvation akin to the fed-bath in simulation 2 and produe the high biomass steady state. Hene, the onditions that lead to the model produing the intermediate biomass steady state are (1) high onentration of gluose in fedbath, and (2) a quik shift of the fed-bath to the ontinuous mode. Further experimentation is, however, neessary to ompletely identify all the model parameters by subjeting the ells to various fed-bathes with a range of substrate onentrations. Follstad and o-workers (1999) have arried out experiments with hybridoma in whih the transition from a low biomass to a high biomass steady state was arried out by dereasing the dilution rate in step hanges to alter the metabolism. The dilution rate was then inreased to reah a more effiient steady state at the original high dilution rate. It is of partiular interest to show that the model presented here is able to vindiate the strategy of Follstad et al. (1999). The agreement is, of ourse, qualitative sine the model was not speifially adapted to their strain and medium. Simulations were arried out with our model, in whih the reator was started as a bath, and shifted to a ontinuous mode with a dilution rate of 0.031/h as before. After the reator reahed steady state, the dilution rate was redued to 0.011/h in steps of 0.01/h. After that, the dilution rate was shifted bak to 0.031/h. The results are depited in Figure 14 and are qualitative agreement in that the reator is indeed shifted to the more desirable steady state. The fluxes alulated from model simulations are depited in Figure 15. This learly demonstrates how a signifiant part of the glyolyti flux through reation la gets hanneled into latate, and this ours in dereasing amounts in the solid, dotted, and dashed steady states. The fluxes S 1 M 3 and S 2 M 7 derease with inreasing biomass onentrations as these states orrespond to dereasing levels of substrate. Also notie that the entry of glutamine into the TCA yle is muh higher in the low biomass steady state whih is in agreement with the experimental results. NAMJOSHI ET AL.: STEADY-STATE MULTIPLICITY IN HYBRIDOMA CULTURES 87

9 step-downs in dilution rates that have also been tried, lead to shifts between the steady states. The multiple steady states for the reator are haraterized by different metaboli states within the ells whih an be understood as different outomes of ompetitions between various enzyme systems. This model although identified for the hybridoma experiments seems to be generally appliable to mammalian systems other than hybridoma (suh as Baby Hamster Kidney (BHK) (Helder et al., 1999) and Chinese Hamster Ovary (CHO) ells) that display similar behavior. Bifuration analysis (Namjoshi and Ramkrishna, 2001) and more dynami simulations are required to ompletely explore the apabilities of the model. Figure 14. Simulations of the yberneti hybridoma model for dilution rate step-up and step-down experiments subsequent to bath startup. CONCLUSIONS The yberneti hybridoma model presented in this work explains multiple steady states in ontinuous reators by apturing the partially substitutable and partially omplementary nature of the two substrates, gluose and glutamine. A promising strategy for the identifiation of yberneti models from experimental data that takes into aount abstrated metaboli networks has been presented. The metaboli network is abstrated in a top-down sense keeping in mind the phenomena being modeled. This strategy uses Metaboli Flux Analysis of experimental data to establish ompetitions within the abstrated metaboli network. The model explains three ontinuous steady states that differ markedly in the level of ell mass (and antibody) and waste metabolites suh as latate. These states are ahieved in ontinuous reators by different start-up strategies like bath and fed-bath. Preliminary simulations have shown that other experimental strategies like step-ups and APPENDIX Presented here are details of the model equations for the bath, fed-bath, and ontinuous ases, as well as the tabulations of the model parameters and initial onditions plus omment upon the solvers used. Model Equations: Speies Balanes Continuous Reator ds 1 dt = r 1av 1a + r 1b v 1b + r m 2 x + D s f 1 s 1 (4) ds 2 dt = r 3v 3 v 3 d + r 7 v 7 d + r 7 m x + D s 2 f s 2 (5) dm 1 dt = Y 1 r 1a v 1a + r 1b v 1b + Y 5 r 5 r 4 v 4 r 6 v 1a Y m1 rˆg r g m 1 (6) dm 2 = Y dt 3 r 3 v 3 v d 3 + Y 4 r 4 v 4 r 5 Y m2 rˆg r g m 2 (7) dm 3 = Y dt 2 r m 2 Y m3 rˆg r g m 3 (8) Figure 15. Model predited fluxes in the three steady states. dm 4 = Y dt 7 r 7 v d 7 + r m 7 Y m4 rˆg r g m 4 (9) dć dt = rˆg r g ć (10) dx dt = r g D x (11) dl dt = Y 6r 6 v 1a x Dl (12) de i dt = r* e + r ei u i r g + e i, i = 1a, 1b, 3, 4, 6, 7 (13) where the yberneti u i variable is one of u 1a, u 1b, u 3u d 3, u 4, u 1a, and u d BIOTECHNOLOGY AND BIOENGINEERING, VOL. 81, NO. 1, JANUARY 5, 2003

10 Bath Reator For a bath reator, the ontinuous reator equations are appliable with dilution rate D 0. Fed-Bath Reator For a fed-bath reator, the volume of the reator hanges as fresh medium is added into the reator. The following are the equations for a fed-bath in whih s 1 as well as s 2 are maintained onstant. f 1 and f 2 are determined to be the flow rates of the two streams supplying s 1 and s 2, respetively. They an be derived by looking at the mass balanes of s 1 and s 2, respetively. Let r s1 and r s2 be the speifi onsumption rates of s 1 and s 2, respetively. The flow rates for media ontaining s 1 and s 2 are given by f 1 and f 2, respetively. r s1 = r 1a v 1a m + r 1b v 1b + r 2 f 2 = r s 2 xv s f 1 s 1 s 2 r s1 xv s f 1 s 2 s f 2 s f 1 s 1 r s2 = r 3 v 3 v 3 d + r 7 v 7 d + r 7 m f 1 = r s 1 xv + s 1 f 2 s 1 f s 1 The speies balanes for the external variables are then modified: r i = r i max p i k i + p i ć ć + ḱ g e i e max, i = 4, 6 (15) r 5 = r 5 max p i k 5 + p i ć ć + ḱ g (16) p r m max i ć i = r im, i = 2, 7 (17) k m i + p i ć + ḱ g p i max r ei = r e i = 1a, 1b, 3, 4, 6, 7 (18) k ei + p i m max 1 m 2 m 3 m 4 rˆg = rˆg (19) m 1 + k g m 2 + k g m 3 + k g m 4 + k g r g = Y 1 r 1a v 1a + r 1b v 1b + y 2 r m 2 + Y 3 r 3 v d 3 + y 7 r 7 v d 7 + r m 7 + y 4 1 r 4 v 4 + y 5 1 r 5 r 6 v 1a + rˆg 1 y m1 y m2 y m3 y m4 (20) where p i is the reatant in the i th reation, whih is s 1 for i 1a, 1b, 2; s 2 for i 3, 7; m 1 for i 4, 6, and m 2 for i 5. The rate of formation of ć and the growth rate are given by r g and r g, respetively. The definitions of yberneti variables are as follows: ds 1 dt = 0, ds 2 dt = 0, dv dt = f 1 + f 2, dl dt = Y 6r 6 v 1a x 1 dv V dt l dx dt = r g 1 dv x, V dt u 1a v 1b r 1a = r 1a + r 1b r 1b u 1b = max r 1a,r 1b r 1b = r 1a + r 1b v 1a r 1a = max r 1a,r 1b On the other hand, if only s 1 is maintained onstant in the fed-bath, the following equations result: ds 1 dt = 0, ds 2 = r g 1 V dt = r s 2 x + 1 dv V dv dt s f 2 s 2, dt x, dl dt = Y 6r 6 v 1a x 1 dv V dt l dv dt = r s 1 xv, dx s f 1 s 1 dt where r s1 r 1a v 1a + r 1b v 1b + r m 2 and r s2 r 3 v 3v d 3 +r 7 v d 7 +r m 7 as before. Kineti Expressions r i = r i max p i k i + p i + p 2 i K i ć ć + ḱ g e i e max, i = 1a, 1b, 3, 7 (14) r 3 r 7 u d 3 = u d r 3 + r 7 = v d 7 r 3 + r 3 = 7 max r 3,r 7 r 7 v d 7 = max r 3,r 7 Y u 3 r 3 Y 3 = u 4 r 4 Y Y 3 r 3 + Y 4 r 4 = v 3 r 3 4 Y 3 r 3 + y 4 r 3 = 4 max Y 3 r 3, Y 4 r 4 Y v 4 r 4 4 = max Y 3 r 3,y 4 r 4 Model Parameters and Steady States The model parameters are listed in Table II. The units used in this paper are grams/liter (g/l) for external variables, and grams/gram dry weight (g/gdw) for all internal variables (inluding enzymes). The unit for time is hours. The k e values are all equal to the orresponding k values, exept for k e1a 20 (k 1a ). Also, r e max g/(gdw h), 0.1/h, r 3 Table II. Parameters in the yberneti hybridoma model. Reation 1a 1b 2m m r max (1/h) k g/l g/l g/l g/l g/gdw g/gdw g/gdw g/l g/l y (g/g) NAMJOSHI ET AL.: STEADY-STATE MULTIPLICITY IN HYBRIDOMA CULTURES 89

11 Table III. Initial onditions and seady-state values. Variable Units IC-1 IC-2 IC-3 SS-1 SS-2 SS-3 s 1 g/l s 2 g/l m 1 g/gdw m 2 g/gdw m 3 g/gdw e 1a g/gdw e 1b g/gdw e 3 g/gdw e 4 g/gdw x g/l L g/l e 6 g/gdw m 4 g/gdw e 7 g/gdw Ć g/gdw V L D 1/h rˆ gmax g/(gdw h), ḱ g 0.01 g/gdw, k g 0.05 g/gdw, r* e 1e 07 g/(gdw h), e max (r* e + r max e )/(rˆgmax + ), Y m1 0.2, Y m2 0.15, Y m3 0.15, Y m4 1 Y m1 Y m2 Y m3 Y m4. For ontinuous operation, a dilution rate of D 0.031/h is used for the low biomass steady state, while D 0.033/h for the intermediate and f f high biomass steady states. Also, s g/l, s g/l. Initial onditions and steady state values are given in Table III. NOMENCLATURE Ć biomass other than intermediates ć onentration of Ć D dilution rate E i ith key enzyme system e i speifi level of the ith key enzyme system e max maximum enzyme level f i volumetri flow rate for the ith stream k i saturation onstant for the ith reation K i substrate inhibition onstant for the ith reation k ei saturation onstant for the expression of the ith key enzyme system ḱ g saturation onstant for intraellular reations on Ć k g saturation onstant for formation of Ć from an intermediate L latate l latate onentration M i ith intermediate m i speifi level of the ith intermediate p i onentration of ith speies r ei enzyme expression rate of the ith key enzyme system r* ei onstitutive rate of the expression of the ith key enzyme system r g growth rate max r g maximum growth rate rˆg rate of formation of Ć max rˆg maximum rate of formation of Ć r i rate of the ith enzyme ontrolled reation m r i rate of the ith reation fully saturated with enzyme max r i maximum value of r i max r im m maximum value of r i r si rate of onsumption of s i S i ith substrate s i onentration of the ith substrate feed onentration of the ith substrate s i f t u i u i d u i v i v i d v i V x Y i Y mi time yberneti variable governing the enzyme expression rate of the ith key enzyme system u i for a onvergent pathway u i for a divergent pathway yberneti variable governing the ativity of the ith key enzyme system v i for a onvergent pathway v i for a divergent pathway volume of the reator biomass onentration yield oeffiient for the ith reation yield oeffiient Greek symbols i i i Referenes maximum enzyme expression rate of the ith key enzyme system turnover rate of the ith key enzyme system maximum rate of the ith reation Baloo S, Ramkrishna D. 1991a. Metaboli regulation in baterial ontinuous ultures: I. Biotehnol Bioeng 38: Baloo S, Ramkrishna D. 1991b. Metaboli regulation in baterial ontinuous ultures: II. Biotehnol Bioeng 38: Batt BC, Kompala DS A strutured kineti modeling framework for the dynamis of hybridoma growth and monolonal antibody prodution in ontinuous suspension ultures. Biotehnol Bioeng 34: Bree MA, Dhurjati P, Geoghegan Jr RF, Robnett BR Kineti modeling of hybridoma ell growth and immunoglobuling prodution in large-sale suspension ulture. Biotehnol Bioeng 32: Cruz HJ, Moreira JL, Carrondo MJT Metaboli shifts by nutrient manipulation in ontinuous ultures of bhk ells. Biotehnol Bioeng 66: Europa AF, Gambhir A, Fu PC, Hu WS Multiple steady states with distint ellular metabolism in ontinuous ulture of mammalian ells. Biotehnol Bioeng 67: Follstad BD, Balarel RR, Stephanopoulous G, Wang DIC Metaboli flux analysis of hybridoma ontinuous ulture steady state multipliity. Biotehnol Bioeng 63: Anshu Gambhir. Personal ommuniation, BIOTECHNOLOGY AND BIOENGINEERING, VOL. 81, NO. 1, JANUARY 5, 2003

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