mammalian cell culture process for monoclonal …...mammalian cell culture process for monoclonal...

Research ArticleMammalian Cell Culture Process for Monoclonal AntibodyProduction Nonlinear Modelling and Parameter Estimation

Dan SeliGteanu1 Dorin S endrescu1 Vlad Georgeanu2 and Monica Roman1

1Department of Automation Electronics and Mechatronics University of Craiova AI Cuza No 13 200585 Craiova Romania2Faculty of General Medicine Carol Davila University of Medicine and Pharmacy Eroilor Sanitari No 8 050474 Bucharest Romania

Correspondence should be addressed to Monica Roman monicaautomationucvro

Received 17 June 2014 Revised 5 December 2014 Accepted 7 December 2014

Academic Editor Lachlan Gray

Copyright copy 2015 Dan Selisteanu et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Monoclonal antibodies (mAbs) are at present one of the fastest growing products of pharmaceutical industry with widespreadapplications in biochemistry biology and medicine The operation of mAbs production processes is predominantly based onempirical knowledge the improvements being achieved by using trial-and-error experiments and precedent practices Thenonlinearity of these processes and the absence of suitable instrumentation require an enhanced modelling effort and modernkinetic parameter estimation strategies The present work is dedicated to nonlinear dynamic modelling and parameter estimationfor a mammalian cell culture process used for mAb production By using a dynamical model of such kind of processes anoptimization-based technique for estimation of kinetic parameters in the model of mammalian cell culture process is developedThe estimation is achieved as a result of minimizing an error function by a particle swarm optimization (PSO) algorithm Theproposed estimation approach is analyzed in this work by using a particular model of mammalian cell culture as a case study butis generic for this class of bioprocessesThe presented case study shows that the proposed parameter estimation technique providesa more accurate simulation of the experimentally observed process behaviour than reported in previous studies

1 Introduction

As the market demand for monoclonal antibodies is increas-ing there is significant interest in developing proper modelsfor mammalian cell culture processes due to the fact thatthese are commonly used as production platforms for mAbswhich are the fastest growing segment of the biopharmaceuti-cal industry [1ndash6] For mAb production various mammaliancell lines are usually exploited such as murine myeloma(NS0) murine hybridomas Chinese hamster ovary (CHO)and PERC6 human cells The selection of expression systemis determined by its capability to deliver high productivitywith suitable product quality attributes [7] Medical applica-tions for mAbs are quite extensive diagnostic tools therapiesfor various cancers rheumatoid arthritis cardiovascularconditions and so on [4 6ndash9]

Typically the industrial operation for mammalian cellculture mAb platforms relies on empirical knowledge [2 310] and the improvements are achieved by using trial-and-error experiments and precedent practices Consequently

process improvements have generally been time-consumingand costly with a high degree of specificity To assist theselaboratory experiments and in practical terms to achievehigh productivity and better quality products it is of obviousinterest to develop model-based applications and to achieveaccurate dynamical models However the specific character-istics of these processes such as complexity nonlinearity andabsence of cheap and reliable instrumentation require anenhanced modelling effort and advanced kinetic parameterestimation strategies

In order to surmount the above-mentioned limitations oftrial-and-error process development the so-called predictivemodels for mammalian cell culture processes are quiteattractive [4] Generically speaking cell culture modellingtechniques are classified based upon whether a dynamic ora pseudo-steady-state interpretation of cellular metabolismis used [2 4 11 12] Being well-known in control systemsthe pseudo-steady-state approach has a biochemical inter-pretation in cell culture processes It is assumed that allmetabolites within the cell culture process are accumulated

Hindawi Publishing CorporationBioMed Research InternationalVolume 2015 Article ID 598721 16 pageshttpdxdoiorg1011552015598721

2 BioMed Research International

or depleted at a rate considerably faster than the overall cellgrowth rate Consequently the concentration of each systemmetabolite and the rate of each metabolic reaction are allconsidered time-invariant [4] This approach is simple andthe obtained models are linear systems which can be easilycomputed regardless of the model size (complexity) Theinformation gathered in such pseudo-steady-state modelsconcerns the metabolic configuration of cell culture How-ever mammalian cells have a complicated internal structurewith several interconnected biochemical processes and withphenomena onmultiple time scalesThus the pseudo-steady-state models cannot describe in detail the changes that occurover a continuous time-horizon (intracellular concentrationprofiles changes in reaction rate due to gene regulation etc)Therefore the dynamic modelling is more appropriate forthese complex (and dynamical) processes In this case asystem of differential equations will describe the bioprocessmodel Inmany cases the difficulty that arises is related to thecomputational problems especially for large and stiff systemsNo matter what modelling method is chosen the complexitytogether with the nonlinearity of these processes is a limitingfactor in model building

In this paper which is an extended work of [13 14] anessential problem in dynamic modelling of cell culture sys-tems is analysed the so-called parameter estimation Themodel of such bioprocesses can be obtained by using dynamicclassical modelling (based on mass balance) or alternativeapproaches such as pseudo-bond-graph method (a versionof bond graph method introduced by Paynter in 1961 andfurther developed in [15ndash26]) However regardless of themodelling method in order to obtain a dynamical modeluseful for process development (including the design ofsome control strategies) the nonmeasurable parameters ofthe mammalian cell culture system must to be estimatedHowever any parameter in a cell culturemodel could [4] havephysical meaning and be measurable by experiment haveclear physical meaning but be experimentally inaccessibleor have no clear physical meaning (eg be purely mathe-matical in nature) Typically optimization-based techniquesare used for the estimation of nonmeasurable parametersof such biological processes [4 27 28] For example aquadratic programming technique was used by Gao et al[27] and a simple discretization scheme combined witha filtered interior point primal-dual line-search algorithm[29] was proposed by Baughman et al [4] Other nonlinearoptimization-based techniques are the genetic algorithmsorthogonal collocation and particle swarm optimizationPSO which have been applied mostly on chemical processes(see eg [30 31]) or in gene regulatory networks modelling[32 33] In order to obtain accurate solutions in the caseof the mAb production process in this paper a particleswarm-based multistep nonlinear optimization algorithm isproposed [34ndash36]

Concerning the applications of PSO for identification ofbiological systems some results were reported for the processof glycerol fermentation by Klebsiella pneumoniae in batchfed-batch and continuous cultures [37ndash41] The estimationapproach used in these works is in most cases a parallelPSO technique which requires a considerable computational

effort Another trend is related to an indirect use of PSOtechnique for estimation more precisely for the training ofa neural network which models the bioprocess [42]

During the last decade PSO algorithms have gainedmuch attention and wide applications in different fields dueto their effectiveness in performing difficult optimizationissues as well as simplicity of implementation and abilityof fast converge to a reasonably good solution PSO is apopulation-based heuristic global optimization techniquefirst introduced by Kennedy and Eberhart [43] and referredto as a swarm-intelligence technique It is motivated fromthe simulation of social behaviour of animals such as birdflocking fish schooling and swarm In this algorithm thepopulation is called a swarm and the trajectory of each parti-cle in the search space is controlled through the medium of aterm called ldquovelocityrdquo according to its own flying experienceand swarm experience in the search space

This paper proposes a multistep PSO version that usestime-varying acceleration coefficients [35] which is devel-oped to solve the nonconvex optimization problem ensuringfast convergence and very good performance Finally theobtained solution is an optimal set of the kinetic parametervalues

The proposed nonlinear modelling and estimationapproaches are analyzed in this work by using a particularmodel of mammalian cell culture as a case study but they aregeneric for this class of bioprocesses A previously publisheddynamic model of mammalian cell culture by Gao et al in[27] is used as a case study More precisely a process of anImmunoglobulin G-secreting murine hybridoma cultured ina growth medium supplemented with proline L-asparagineand L-aspartic acid is taken into consideration

2 Methods

21 MAb Synthesis by Mammalian Cell Culture ProcessDescription and Modelling Issues In order to model themammalian cell culture processes first it is necessary to ana-lyze the reconstruction of metabolic activities However thereconstruction generally includes only a subset of the highlyactivemetabolic units found in proliferatingmammalian cells[4 34] After the choice of this key subset the next stepconsists in the modelling of the reactionsrsquo rates in the frameof reconstruction This process is a very difficult one andthe modelling of reactions as single-enzyme processes byusing in vitro kinetic parameters is possible only for simpleand small reconstructions Often in vitro kinetic parametersdo not compare well against in vivo observations [4] Thereconstruction in complex processes will frequently combinea number of discrete processes into a single lumped processand will then apply kinetic parameters to the lumped processConsequently some kinetic parameters that appear in themodel may have small or no physical significance and usuallytheir values are not experimentally measurable [4]

Therefore if all of the interaction of metabolites andcell physiology are included in the modelling process thenthe size of the obtained model is very large and it is notappropriate for model-based optimization and control pur-poses The usual solution is to select a priori the elementary

BioMed Research International 3

Table 1 Macroreactions of the mAb production process [4 27]

Reaction number Macroreaction scheme1 GLC rarr 2LAC2 GLC + 2GLU rarr 2ALA + 2LAC3 GLC + 2GLU rarr 2ASP + 2LAC4 GLU rarr PRO5 ASN rarr ASP + NH3

6 GLN + ASP rarr ASN + GLU7 00508GLC + 00577GLN + 00133ALA + 0006ASN + 00201ASP + 00016GLU + 0081PRO rarr BM8 00104GLN + 0011ALA + 0072ASN + 0082ASP + 00107GLU + 00148PRO rarr MAb9 GLN rarr GLU + NH3

reaction schemes and to relate themajor macroscopic speciessuch as biomass essential substrates and products by a setof so-called macroreactions [27] Thus a simplified modelis obtained which is suitable for optimization and controlAs was mentioned before the next step in the modelling isrelated to the determination of reaction kinetics and the finalmodel is obtained based on mass balance equations of themacroscopic species involved in the reactions

Next a particular model of mammalian cell culturepublished by Gao et al [27] will be described and used asa case study Gao et al [27] provided a detailed description ofan Immunoglobulin G- (IgG-) secreting murine hybridoma(130-8F Sanofi Pasteur) cultured in a D-MEM (DulbeccorsquosModified Eagle Medium) growth medium supplementedwith proline L-asparagine and L-aspartic acid In this pro-cess batch cultures of the organismwere allowed to grow for aminimum of 7 days The infrequent measured concentrationdata for glucose lactate and ammonia as well as for 20 aminoacids and the monoclonal antibody were obtained from thecollected samples via proper techniques By using the mea-sured data the average rates of transmembrane fluxes werecalculated for eachmetabolite for both the initial exponentialgrowth phase and for the postexponential (decline) phaseGao et al [27] used themetabolic flux analysis (MFA) in orderto calculate the unknown intracellular fluxes from measuredextracellular fluxes by applying steady-state mass balanceequations The obtained metabolic network was constructedbased on some preliminary studies [44ndash47] and it representsthe significant metabolic pathways in proliferating animalcells Gao et al [27] determined that 16 reactions (a halfof the total number) in the chosen reconstruction did notfunction significantly and consequently these reactions withan activity of about 1 of the total were eliminated Theremaining subset of 16 reactions of the reduced metabolicreconstruction was further reduced by using a techniquethat combines reactions that share commonmetabolites [48]Finally the reduced reaction scheme for this mAb bioprocesscontains a number of only 11 extracellular compounds and itconsists of nine macroreactions presented in Table 1 [4 27]

The dynamical model of a generic bioprocess inside abioreactor can be obtained by using the mass balance of

the component and it is given by the following set of differ-ential equations [49]

119889120585

119889119905

= 119870 sdot 120593 (120585) + 119863 sdot 120585 + 119865 minus 119876 (1)

where 120585 = [1205851

1205852

sdot sdot sdot 120585119899]119879 is the 119899-dimensional vec-

tor of the instantaneous concentrations (the concentrationsof extracellular metabolites in our particular case) 120593 =

[1205931

1205932

sdot sdot sdot 120593119898]119879 is the vector of the reaction rates and

119870 is the 119899times119898 dimensional matrix of stoichiometric (or yield)coefficients with 119870 = [119870

119894119895] 119894 = 1 119899 119895 = 1119898 where

119870119894119895= (plusmn)119896

119894119895if 119895 sim 119894 The notation 119895 sim 119894 indicates that the sum

is done in accordance with the reactions 119895 that involve thecomponents 119894 The sign of the yield coefficients 119896

119894119895is given by

the type of the component 120585119894 plus (+) when the component is

a reaction product and minus (minus) otherwise119863 is the specificvolumetric outflow rate (hminus1) usually called dilution rate In(1) 119865 = [119865

11198652

sdot sdot sdot 119865119899]119879 is the vector of rates of liquid

supply and 119876 = [1198761

1198762

sdot sdot sdot 119876119899]119879 is the vector of rates

of removal of the components in gaseous formModel (1) describes in fact the behaviour of an entire class

of bioprocesses and is referred to as the general dynamicalstate-space model of this class [49 50] In (1) the term119870 sdot 120593(120585) is in fact the rate of consumption andor productionof the components in the reactor that is the reaction kineticsThe term minus119863120585 + 119865 minus 119876 represents the exchange with theenvironment The strongly nonlinear character of this modelis given by the reaction kinetics In many practical situationsthe structure and the parameters of the reaction rates arepartially known or even completely unknown

Typically in a batch process the reactor is filled with thereactant mixture substrates and microorganisms Then thereactions occur inside the reactor for a time period afterthat the products are removed from the tank Because thestudied bioprocess takes place inside a batch reactor model(1) becomes

119889120585

119889119905

= 119870 sdot 120593 (120585) (2)

that is the term minus119863120585+119865minus119876 (which represents the exchangewith the environment) is zero in this particular batch mode


For the mAb production process the concentrations ofthe 11 extracellular metabolites (given in the reaction schemefrom Table 1) constitute the elements of the state vector frommodel (1) and are denoted as follows

120585 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1205851

1205852

1205853

1205854

1205855

1205856

1205857

1205858

1205859

12058510

12058511

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198781

1198782

1198783

1198784

1198785

1198751

1198752

1198753

1198754

1198755

1198756

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

GLCGLNGLUASNASPLACALAPROMAbBMNH3

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(3)

where GLC = glucose GLN = glutamine GLU = glutamateASN = asparagine ASP = aspartate LAC = lactate ALA =alanine PRO = proline MAb = monoclonal antibody BM =biomass NH

3= ammonia are themetabolites given in Table 1

(and for simplicity the concentrations of the correspondingelements in model (1))

However in order to complete the model of the mAbproduction process it is necessary to add the evolution ofthe viable cell concentrations of the culture because themetabolite mass balances depend on the amount of viablecells Gao et al [27] noticed the typical behaviour of thebatch culture with exponential growth and postexponentialdecline senescence phase (which occurs after the first phaseof evolution due to the aging of the cells and the accu-mulation of autoinhibitory metabolites) Therefore anothertwo concentrations enter in the complex model of thebioprocess the viable cell concentration 119883 and the dead cellconcentration119883

119889 The dynamics of these concentrations will

be modelled separately depending of the phase (growth ordecay)

Remark 1 To be exact for the mAb production process theexchange with environment is zero except the CO

2gaseous

flow but this flow is not measured and CO2is not predicted

in the final model as it is considered in [27]

In the following the dynamical model (2) of the mAbproduction process will be presented starting with thereaction scheme given in Table 1 Afterward the problemof kinetic rates is addressed together with the parameterestimation problem via PSO-based techniques

The dynamical model of the form (2) can be particu-larized for the mAb production process described by thereaction scheme from Table 1 by using the mass balanceof the components (via classical methods [4 27] or bondgraph approach [13]) inside the batch reactor The followingdynamical model is obtained

1198891198781

119889119905

= minus 1205931minus 1205932minus 1205933minus 119896171205937

1198891198782

119889119905


1198891198783

119889119905

= minus 119896321205932minus 119896331205933minus 1205934+ 1205936minus 119896371205937minus 119896381205938+ 1205939

1198891198784

119889119905

= minus 1205935+ 1205936minus 119896471205937minus 119896481205938

1198891198785

119889119905

= 119896531205933+ 1205935minus 1205936minus 119896571205937minus 119896581205938

1198891198751

119889119905

= 119896611205931+ 119896621205932+ 119896631205933

1198891198752

119889119905

= 119896721205932minus 119896771205937minus 119896781205938

1198891198753

119889119905

= 1205934minus 119896871205937minus 119896881205938

1198891198754

119889119905

= 1205938

1198891198755

119889119905

= 1205937

1198891198756

119889119905

= 1205935+ 1205939

(4)

Model (4) can be written in a compact form [13]

119889

119889119905

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198781

1198782

1198783

1198784

1198785

1198751

1198752

1198753

1198754

1198755

1198756

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟

120585

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus1 minus1 minus1 0 0 0 minus11989617

0 0

0 0 0 0 0 minus11989626

minus11989627

minus11989628

minus1

0 minus11989632

minus11989633

minus1 0 1 minus11989637

minus11989638

1

0 0 0 0 minus1 1 minus11989647

minus11989648

0

0 0 11989653

0 1 minus1 minus11989657

minus11989658

0

11989661

11989662

11989663

0 0 0 0 0 0

0 11989672

0 0 0 0 minus11989677

minus11989678

0

0 0 0 1 0 0 minus11989687

minus11989688

0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 1 0 0

0 0 0 0 1 0 0 0 1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119870

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1205931

1205932

1205933

1205934

1205935

1205936

1205937

1205938

1205939

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟

120593(120585)

(5)


where the values of stoichiometric coefficients are given inthe reaction schemes from Table 1 and are as follows [4 27]11989617

= 00508 11989626

= 1 11989627

= 00577 11989628

= 00104 11989632

= 211989633

= 2 11989637

= 00016 11989638

= 00107 11989647

= 0006 11989648

=

0072 11989653

= 2 11989657

= 00201 11989658

= 0082 11989661

= 2 11989662

= 211989663

= 2 11989672

= 2 11989677

= 00133 11989678

= 0011 11989687

= 0081and 11989688

= 00148The nonlinear dynamical model (5) is obvious from

the general form (2) However in order to complete themodel of mAb production process it is necessary to addthe submodels corresponding to the dynamics of viable cellconcentration and dead cell concentration respectively Hereit should be noted that Gao et al [27] have obtained from theexperimental observations that the model describing viablecell growth changes at 119905 = 54 h to reflect the transition fromexponential growth to the decline phase With this remarkthe dynamical model of the viable and cell concentrationsevolutions is as follows [27]

119889119883

119889119905

= 120583119883 minus 119896119889119883119883119889 for 119905 lt 119905exp

119889119883

119889119905

= minus 119896119889119883119883119889 for 119905 ge 119905exp

119889119883119889

119889119905

= 119896119889119883119883119889

(6)

where 120583 is the specific growth rate of the viable cells 119896119889is a

kinetic (decay) parameter and 119905exp is the time period of theexponential growth phase

The most difficult modelling problem for the systemof differential equations (5) (6) is related to the model ofnonlinear reaction kinetics Gao et al [27] suggested thata generalized form of saturable kinetics (ie compoundMonod kinetics) is suitable to describe the rate of eachmacroreaction from the reaction scheme given in Table 1Rates for each of these macroreactions were expressed in thenext compact form [4 31]

120593119894= 120593

lowast

119894sdot 119883 sdot prod

119878119895isin119878119894

119878119895

119870119878119895119894

+ 119878119895

119894 = 1 9 (7)

In the kinetic rates expression (7) 120593119894is the reaction

rate for reaction 119894 120593lowast119894is the maximum reaction rate for

reaction 119894 119878119895is the concentration of substrate 119895 within the

set 119878119894of substrates for reaction 119894 and 119870

119878119895119894is a kinetic half-

saturation constant for substrate 119895 in reaction 119894 The specificrate expressions for each macroreaction are given in Table 2[4 27] As Baughman et al [4] noticed the rate expressionsfor macroreactions 7 and 8 do not rigorously conform tothe general format (7) More precisely it was assumed thatthe principal rate-limiting substrate for both biomass andantibody synthesis is glutamine and the kinetic contributionsof any other substrates were thus omitted

In conclusion the full dynamical model of mAb produc-tion process is given by (5) where the kinetic rates are ofthe form presented in Table 3 together with the dynamicalmodels (6) of viable and dead cell evolution in the batchreactor

Table 2 Kinetics expressions for the macroreactions [4 27]

Reaction number Kinetic rate

1 1205931= 120593lowast

1119883

1198781

11987011987811

+ 1198781

2 1205932= 120593lowast

2119883

1198781

11987011987812

+ 1198781

1198783

11987011987832

+ 1198783

3 1205933= 120593lowast

3119883

1198781

11987011987813

+ 1198781

1198783

11987011987833

+ 1198783

4 1205934= 120593lowast

4119883

1198783

11987011987834

+ 1198783

5 1205935= 120593lowast

5119883

1198784

11987011987845

+ 1198784

6 1205936= 120593lowast

6119883

1198782

11987011987826

+ 1198782

1198785

11987011987856

+ 1198785

7 1205937= 120593lowast

7119883

1198782

11987011987827

+ 1198782

8 1205938= 120593lowast

8119883

1198782

11987011987828

+ 1198782

9 1205939= 120593lowast

9119883

1198782

1198701198789+ 1198782

The state variables within the dynamical model (5)ndash(7) are associated with components of the macroreactionsfrom the reaction scheme given in Table 1 While thesecomponents represent biological variables (concentrations ofsome substances or compounds) the kinetic parameters donot have always clear measurable physical representations

The problem that remains to be solved now is relatedto the estimation of the unknown (inaccessible) kineticparameters of the dynamical model (5) (6) of the mam-malian cell culture Therefore it is necessary to estimate theexperimentally inaccessible parameter values for the modelthat provide the best approximation to the measured cultureconcentrations data

22 PSO-Based Technique Parameter Estimation

221 Problem Statement and Basic PSO Algorithms At thebeginning of parameter estimation the input and output dataare known and the real system parameters are assumed asunknown The identification problem is formulated in termsof an optimization problem in which the error between anactual physical measured response of the system and thesimulated response of a parameterized model is minimizedThe estimation of the systemparameters is achieved as a resultof minimizing the error function by the PSO algorithm

Consider that the nonlinear system (2) that describes thedynamical behaviour of a class of bioprocesses is written asthe following 119899-dimensional nonlinear system

119889120585

119889119905

= 119870 sdot 120593 (120585) = 119891 (120585 119905 120579) (8)

where 120585 isin R119899 is the state vector (ie the vector ofconcentrations) 120579 isin R119902 is the unknown parameters vector(ie the vector of unknown kinetic parameters) and 119891 is agiven nonlinear vector function


Table 3 Experimental concentration measurements [4 27]

Time 0 h 28 h 54 h 76 h 101 h 124 h 147 hGLC [mM] 359 plusmn 004 259 plusmn 009 188 plusmn 015 177 plusmn 005 170 plusmn 003 167 plusmn 004 168 plusmn 005GLN [mM] 285 plusmn 004 127 plusmn 030 042 plusmn 031 011 plusmn 010 000 plusmn 000 000 plusmn 000 000 plusmn 000ASN [mM] 046 plusmn 000 039 plusmn 002 035 plusmn 002 032 plusmn 003 028 plusmn 002 025 plusmn 002 022 plusmn 002ASP [mM] 027 plusmn 002 018 plusmn 002 010 plusmn 004 007 plusmn 004 004 plusmn 004 003 plusmn 004 003 plusmn 004LAC [mM] 051 plusmn 001 133 plusmn 006 166 plusmn 019 174 plusmn 001 171 plusmn 001 172 plusmn 001 173 plusmn 005ALA [mM] 033 plusmn 003 072 plusmn 011 115 plusmn 017 132 plusmn 012 146 plusmn 007 148 plusmn 007 151 plusmn 008PRO [mM] 030 plusmn 001 027 plusmn 002 042 plusmn 004 053 plusmn 002 056 plusmn 002 060 plusmn 001 060 plusmn 001MAB [10minus4mM] 034 plusmn 012 102 plusmn 006 158 plusmn 016 231 plusmn 024 266 plusmn 041 309 plusmn 060 341 plusmn 075BM [mM] 201 plusmn 020 1161 plusmn 046 1651 plusmn 085 1798 plusmn 084 1941 plusmn 221 1867 plusmn 249 1797 plusmn 133119883 [106 cellsmL] 009 plusmn 001 058 plusmn 002 079 plusmn 005 072 plusmn 001 047 plusmn 006 017 plusmn 003 006 plusmn 002119883119889[106 cellsmL] 002 plusmn 001 005 plusmn 001 011 plusmn 001 025 plusmn 005 058 plusmn 005 085 plusmn 012 091 plusmn 007

To estimate the unknown parameters in (8) a parameteridentification system is defined as follows

119889120585 (119905)

119889119905

= 119891 (120585 119905

120579)

(9)

where 120585 isin R119899 is the estimated state vector and 120579 isin R119902 is theestimated parameters vector

Theobjective function defined as themean squared errorsbetween real and estimated responses for a number 119873 ofgiven samples is considered as fitness of estimated modelparameters [14]

119882 =

1

119873 +119872

119872

sum

119895=1

119873

sum

119896=1

(120585

119896

119895minus120585

119896

119895)

2

(10)

where 119872 is the number of measurable states and 119873 is thedata length used for parameter identification whereas 120585119896

119895and

120585119896

119895are the real and estimated values of state 119895 at time 119896

respectivelyThis objective function is a function difficult to minimize

because there are many local minima and the global mini-mum has a very narrow domain of attraction Our goal isto determine the system parameters using particle swarmoptimization algorithms in such a way that the value of 119882is minimized approaching zero as much as possible

Mathematical description of basic PSO and some impor-tant variants is presented in the following

Candidate solutions of a population called particles coex-ist and evolve simultaneously based on knowledge sharingwith neighbouring particles Each particle represents a poten-tial solution to the optimization problem and it has a fitnessvalue decided by optimal function Supposing search spaceis 119872-dimensional each individual is treated as a particlein the 119872-dimensional search space The position and rateof position change for 119894th particle can be represented by119872-dimensional vector 119909

119894= (119909

1198941 1199091198942 119909

119894119872) and V

119894=

(V1198941 V1198942 V

119894119872) respectively The best position previously

visited by the 119894th particle is recorded and represented by119901119894= (1199011198941 1199011198942 119901

119894119872) called 119901119887119890119904119905 The swarm best position

previously visited by all the particles in the populationis represented by 119901

119892= (119901

1198921 1199011198922 119901

119892119872) called 119892119887119890119904119905

Then particles search their best position which are guidedby swarm information 119901

119892and their own information 119901

119894

Each particle modifies its velocity to find a better solution(position) by applying its own flying experience (ie memoryof the best position found in earlier flights) and the experi-ence of neighbouring particles (ie the best solution foundby the population) Each particle position is evaluated byusing fitness function and updates its position and velocityaccording to the following equations

V119896+1119894

= 120596 sdot V119896119894+ 11988811199031(119901119887119890119904119905

119896

119894minus 119909

119905

119894) + 11988821199032(119892119887119890119904119905

119896

119894minus 119909

119896

119894)

119909

119896+1

119894= 119909

119896

119894+ V119896+1119894

(11)

where 119896 is iteration number 120596 is inertia weight 1198881and 1198882are

two acceleration coefficients regulating the relative velocitytoward local and global best position and 119903

1and 119903

2are

two random numbers from the interval [0 1] Many effectshave been made over the last decade to determinate theinertia weight Various studies have shown that under certainconditions convergence is guaranteed to a stable equilibriumpoint [51] These conditions include 120596 gt (119888

1+ 1198882)2 minus 1 and

0 lt 120596 lt 1 The technique originally proposed was to boundvelocities so that each component of V

119894is kept within the

range [119881min 119881max]Unfortunately this simple form of PSO suffers from

the premature convergence problem which is particularlytrue in complex problems since the interacted informationamong particles in PSO is too simple to encourage a globalsearch Many efforts have been made to avoid the prematureconvergence One solution is the use of a constriction factorto insure convergence of the PSO introduced in [45] Thusthe expression for velocity has been modified as

V119896+1119894

= ℎ sdot [V119896119894+ 11988811199031(119901119887119890119904119905

119896

119894minus 119909

119905

119894) + 11988821199032(119892119887119890119904119905

119896

119894minus 119909

119896

119894)]

119909

119896+1

119894= 119909

119896

119894+ V119896+1119894

(12)

where ℎ represents the constriction factor and is defined as

ℎ =

2

(

100381610038161003816100381610038162 minus 120572 minus radic120572

2minus 4120572

10038161003816100381610038161003816)

120572 = 1198881+ 1198882gt 4 (13)


In this variant of the PSO algorithm ℎ controls the mag-nitude of the particle velocity and can be seen as a dampeningfactor It provides the algorithm with two important features[52] First it usually leads to faster convergence than standardPSO Second the swarm maintains the ability to performwide movements in the search space even if convergence isalready advanced but a new optimum is found Thereforethe constriction PSO has the potential to avoid being trappedin local optima while possessing a fast convergence It wasshown to have superior performance compared to a standardPSO [53]

It is shown that a larger inertia weight tends to facilitatethe global exploration and a smaller inertia weight achievesthe local exploration to fine-tune the current search areaThebest performance could be obtained by initially setting 120596 tosome relatively high value (eg 09) which corresponds toa system where particles perform extensive exploration andgradually reducing 120596 to a much lower value (eg 04) wherethe system would be more dissipative and exploitative andwould be better at homing into local optima In [54] a linearlydecreased inertia weight 120596 over time is proposed where 120596 isgiven by the following equation

120596 = (120596119894minus 120596119891) sdot

(119896max minus 119896)

119896max+ 120596119891 (14)

where 120596119894 120596119891are starting and final values of inertia weight

respectively 119896max is the maximum number of the iterationand 119896 is the current iteration number It is generally taken thatstarting value is 120596

119894= 09 and final value is 120596

119891= 04

On the other hand in [35] PSOwas introducedwith time-varying acceleration coefficients The improvement has thesame motivation and similar techniques as the adaptationof inertia weight In this case the cognitive coefficient 119888

1is

decreased linearly and the social coefficient 1198882is increased

linearly over time as follows

1198881= (1198881119891minus 1198881119894) sdot

(119896max minus 119896)

119896max+ 1198881119894

1198882= (1198882119891minus 1198882119894) sdot

(119896max minus 119896)

119896max+ 1198882119894

(15)

where 1198881119894

and 1198882119894

are the initial values of the accelerationcoefficients 119888

1and 1198882and 1198881119891

and 1198882119891

are the final values ofthe acceleration coefficients 119888

1and 1198882 respectively Usually

1198881119894= 25 119888

2119894= 05 119888

1119891= 05 and 119888

2119891= 25 [14 35 36]

The dynamical model of mAb production process givenby the relations (5) (6) associated with the expressions ofthe kinetic rates presented in Table 3 contains a number of 23kinetic parameters (maximum reaction rates and kinetic half-saturation constants) In order to estimate these unknown(inaccessible) kinetic parameters of the complex dynamicalmodel of mammalian cell culture the measured concentra-tions are used and a PSO-based algorithm is implementedThe goal is to obtain a model that approximates as well aspossible the behaviour of the process (expressed by meansof the experimentally obtained data) The model of mAbproduction process under investigation is in fact based on

several macroreactions therefore it results in the fact that thekinetic parameters do not have always clearmeasurable phys-ical representations Thus an optimization-based estimationtechnique is suitable for this set of kinetic parameters

222 Implementation of PSO-Based Technique In the follow-ing a multistep PSO-based version that uses time-varyingacceleration coefficients is implemented and an optimal setof kinetic parameter values of the mAb production process isobtained

In order to implement the PSO-based technique themodel of mAb production process (5) (6) obtained from themacroreactions schemes is used translated into the genericparameter identification system represented in (9)

The experimental concentration values for all theinvolved extracellular metabolites are provided in the workof Gao et al [27] The batch cultures of the organism wereallowed to grow for 147 h with 54 h the exponential growthphase and 93 h the postexponential (decline) phase Theinfrequent measured concentration data for the metaboliteswere obtained from the collected samples via propertechniques The set of concentrations measurements aregiven in Table 3 [4 27] Each data point is the average ofmeasurements taken from three independent experimentswith standard deviation [4 27]

To facilitate the application of the proposed PSO-basedparameter estimation strategy the time derivatives of thestates from model (5)ndash(7) must be reconstructed Becausethe measured data are very few (only 7 experimental mea-surements for each parameter see Table 3) an interpolationmethod is necessary to find intermediate values of the stateswhich are actually the biological parameters of the processthat is the concentrations ofmetabolites Such situationswitha small number of experimental measurements are typicalfor many bioprocesses Ideally speaking the online mea-surements (in each sampling moment at every 6min eg)for each concentration are necessary However these onlinemeasurements are achieved with expensive instrumentationor there are no such fast sensors for some concentrationsThus the infrequent offline measurements are preferredTo facilitate the achievement of an accurate estimation ofmodel parameters of mAb process we need the interpolationof these measured data which allows us to obtain theunavailable data between adjacent measurement points (ieto estimate the unavailable data needed to calculate modelpredictions between these measurement points)

Remark 2 From mathematical point of view a discussionabout the interpolation technique can be done Many authorsuse the linear interpolation with advantages related torapidity and simple implementation [4] However the linearinterpolation is not very precise Another disadvantage isthat the interpolant is not differentiable at the points wherethe value of the function is known Therefore we propose acubic interpolation method that is the simplest method thatoffers true continuity between the measured data A cubicHermite spline or cubicHermite interpolator is a splinewhereeach piece is a third-degree polynomial specified in Hermiteform that is by its values and first derivatives at the end


End

Yes No

Calculate fitness values for particles

Is current fitness value better than

Keep previous Assign current fitness

Start

Assign best particlersquos

Update each particleposition

Calculate velocity foreach particle

Target or maximum epochs reached

No

No

k = 1

Initialize particles for problem Pk

k = k + 1

pbest

pbestas new pbest

pbest value to gbest

Yes

Yesk le 9

Figure 1 The flowchart of the multistep PSO algorithm

points of the corresponding domain interval Cubic Hermitesplines are typically used for interpolation of numeric dataspecified at given argument values 119905

1 1199052 119905

119899 to obtain a

smooth continuous functionTheHermite formula is appliedto each interval (119905

119896 119905119896+1

) separately The resulting spline willbe continuous and will have continuous first derivative

The time derivatives of the states are approximated usingforward differences

119889120585 (119905119896)

119889119905

asymp

120585 (119905119896) minus 120585 (119905

119896+ 119879119904)

119879119904

(16)

where 119879119904represents the sampling period In this approxima-

tion the error is proportional with the sampling interval (asmaller sampling period will give a smaller approximationerror)

Because a 23-dimensional optimization problem thatmust be solved for simultaneously estimation of all unknownparameters requires great computational resources a mul-tistep approach was used So the problem was split innine simpler problems that are solved sequentially until all23 parameters are found These problems are noted with1198751 1198752 1198759 and the corresponding resulted parametersare presented in Table 4 A flowchart of the multistep PSOalgorithm is given in Figure 1

Table 4The subproblems solved by using the multistep PSO-basedapproach

Subproblem ParametersP1 120593

lowast

7 11987011987827

P2 120593lowast

8 11987011987828

P3 120593lowast

4 11987011987834

P4 120593lowast

2 11987011987812 11987011987832

P5 120593lowast

6 11987011987826 11987011987829 120593lowast

9 11987011987856

P6 120593lowast

5 11987011987845

P7 120593lowast

3 11987011987813 11987011987833

P8 120593lowast

1 11987011987811

P9 120583 119896119889

For example the problem 1198751 corresponds to the 10thequation from system (5)-(6) (that represents time evolutionof the biomass) and only two parametersmust be estimated inthis case120593lowast

7and119870

11987827ThePSO algorithm is used tominimize

the sumof the square errors betweenmeasured and estimateddata

1198821198751=

119873

sum

119896=1

(12058510(119896 sdot 119879119904) minus

12058510(119896 sdot 119879119904))

2


Table 5Kinetic parameter estimates obtained via themultistepPSOapproach

Kinetic parameters Estimated values120593lowast

1[pmol(cell h)] 8443 times 10minus4

120593lowast

2[pmol(cell h)] 2481 times 106

120593lowast


120593lowast


120593lowast

5[pmol(cell h)] 7283

120593lowast


120593lowast


120593lowast


120593lowast


11987011987811

[mM] 8989 times 105

11987011987812

[mM] 6495 times 104

11987011987813

[mM] 3723 times 104

11987011987832

[mM] 7076 times 102

11987011987833

[mM] 2782 times 103

11987011987834

[mM] 001911987011987826

[mM] 2719 times 104

11987011987827

[mM] 9324 times 103

11987011987828

[mM] 053711987011987829

[mM] 6683 times 105

11987011987845

[mM] 1920 times 103

11987011987856

[mM] 4488 times 104

120583 0043119896119889

0067

st 12058510((119896 + 1) sdot 119879

119904)

=12058510(119896 sdot 119879119904) + 119879119904sdot

120593lowast

7sdot 1205852(119896 sdot 119879119904) sdot 12058512(119896 sdot 119879119904)

11987011987827

+ 1205852(119896 sdot 119879119904)

(17)

3 Results and Discussion

31 Optimal Set of Kinetic Parameters The optimizationproblem formulated in the previous section is nonlinearand nonconvex with many local minima The estimatedparameters of one subproblem are then considered knownin the subsequent equations In order to be clear the alreadyestimated parameters are not updated between solutions ofsubproblems For example in the frame of problem 1198751 twoparameters are estimated 120593lowast

7and119870

11987827 These parameters will

be available in the next problem (1198752) and so on In this studyall the computations were achieved with a sampling period119879119904= 6min (01 h) As example for problem 1198751 a number of

150 particles randomly initialized were used The algorithmstops if the square error is smaller then 10119890 minus 6 or after 300iterations The optimal set of kinetic parameter values of themAb production process obtained via this multistep PSO-based approach is given in Table 5

The partition of themultidimensional optimization prob-lem proposed within our PSO algorithm not only ensures the

0 50 100 15002468

10121416182022

Time (h)

Biom

ass (

mM

)

PSO parameter estimationBaughman et al [4]

Gao et al [27]Measured concentrations

Figure 2 Simulation results profile of the biomass concentration

decrease of necessary computational resources by compari-son with Gao et al [27] and Baughman et al [4] approachesbut additionally offers a solution for the reported problemsconcerning the stiffness of estimated parameter set Moreprecisely as Baughman et al [4] noticed there are someconcentrations such as the mAb concentration that areseveral orders of magnitude less than other metabolites Thisfact leads to stiffness problems in the optimization procedurewhich are partially solved in [4] by using an alternativeerror objective for the mAb concentration In our approachthe partition in simpler PSO problems solves uniformly thisissue using the same error objective for the entire set ofparameters

Another important problem approached and solved byusing the proposed PSO method is related to the expressionsof reaction kinetics To simplify the optimization problemGao et al [27] considered that the values of half-saturationconstants are sufficiently small and consequently the kineticrates given in Table 2 can be simplified such that the relation(7) becomes120593

119894= 120593lowast

119894sdot119883 119894 = 1 9This simplification eliminates

the necessity of half-saturation constants estimation andonly the maximum reaction rates need to be estimatedHowever as was mentioned in [4] half-saturation constantscan be significantly smaller than the corresponding substrateconcentration in processes controlled by a single enzyme(eg glucose transport) but this fact is not necessarilytrue for the macroreactions in mAb production processesTherefore Gao et al assumption is unwarranted and it canaffect the reliability of the model This is one of the reasonsbecause the proposed PSO method yield good fitting resultscomparedwithGao et al [27] as it can be seen in Figures 2ndash5

Remark 3 There are necessary some comments concerningthe overfitting problems Overfitting arises when a statisticalmodel describes noise instead of the underlying relationshipit usually occurs when a model is very complex such as


0 50 100 15015

2

25

3

35

4

Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)

Glu

cose

(mM

)

0

05

1

15

2

25

3

35

Glu

tam

ine (

mM

)

01

02

03

04

05

06

Asp

arag

ine (

mM

)

0

005

01

015

02

025

03

Asp

arta

te (m

M)

0 50 100 150Time (h)





Figure 3 Simulation results profiles of substrate concentrations GLC GLN ASN and ASP

having many parameters relative to the number of observa-tions Even though the approached mAb production modelis quite complex it is not a statistical model Also even ifthe number of measured concentration samples is relativelysmall by comparisonwith the number of parameters the PSOtechnique is an optimization procedure which is much lesssensitive to overfitting than the methods that are based onmodel training such as neural network techniques (see Tetkoet al [55]) The potential for overfitting depends not only onthe number of parameters and measured data but also on theconformability of the model structure with the data shape

Some comparisons of the proposed PSO approach withother PSO applications to bioprocesses can be done Most ofthe reported works were focused on the process of glycerolfermentation by Klebsiella pneumoniae in batch fed-batchand continuous cultures [37ndash41] Shen et al [37] studieda mathematical model of Klebsiella pneumoniae in a con-tinuous culture An eight-dimensional nonlinear dynamicalsystemwas obtained and a parallel PSO techniquewas imple-mented in order to identify 19 parameters The identificationresults are compared only with experimental steady-state

values The reported mean relative errors between the com-putational values and the experimental data are quite large(between 8 and 13) A similar model of bio-dissimilationof glycerol by K pneumoniae in a continuous culture waswidely analyzed by Zhai et al in [38] Here a parallel PSOpathways identification algorithmwas constructed to find theoptimal pathway and 21 parameters under various conditionsThe combined estimation of pathways and process parame-ters leads to a vast identification model solved on a clusterserver with 16 nodes (each node with 4 Core 64-bit 25 GHzprocessor) in over 130 hours Comparable PSO approacheswere used in [39 41] in the case of the same fermentationprocess but in a batch culture For example Yuan et al[39] used a parallel migration PSO algorithm to estimatepathways and 12 parameters of the eight-dimensional modelThe identification problem was split into two subproblems(one for pathways and one for parameters) and solved onthe above cluster in about 18 hours Another work addressedthe PSO identification in the case of the fermentation ofglycerol byK pneumoniae in a fed-batch culture [40] A non-linear hybrid system was developed (with seven differentialequations plus a switching mechanism and 8 parameters)


50 100 150

05

1

15

2

25

Time (h)

Lact

ate (

mM

)

04

06

08

1

12

14

16

18

2

Ala

nine

(mM

)

03

04

05

06

07

Prol

ine (

mM

)

0

1

2

3

4

0 50 100 150Time (h)

0

50 100 150Time (h)

050 100 150Time (h)

0

times10minus4





Ant

ibod

y (10eminus4

mM

)

Figure 4 Simulation results profiles of concentrations LAC PRO ALA and MAb

The proposed technique was an asynchronous parallel PSOand the reported averaged computational time was about326 h on the above-mentioned cluster server with 16 nodesThe reported results in [38ndash40] are very good even theaccuracy of the algorithms cannot be fully assessed (statisticalreports were not provided) However the computationaleffort is considerable given the fact that simultaneous path-ways and parameters identification was approached

The particle swarm-based multistep nonlinear optimiza-tion algorithm proposed in the present work was used for theestimation of 23 parameters of an eleven-dimensional nonlin-ear system (the pathways identification was not considered)By using themultistep approach the computational effortwasquite small (about 20min on a computer with Intel Corei5 64-bit 33 GHz processor) The obtained results and thestatistical analysis show a good accuracy of the identificationresults

32 Simulation Results The performance of the proposedestimation technique was analyzed by using numerical sim-ulations All these simulations are achieved by using thedevelopment programming and simulation environment

MATLAB (registered trademark of The MathWorks IncUSA) For comparison the simulated profiles based on thekinetic parameters obtained via PSO technique (Table 5) arerepresented together with the original system measurements[27] and with the profiles obtained by Gao et al [27] andBaughman et al [4] respectively The concentration profilesbased on the results of Gao et al and on the results ofBaughman et al [4] approach respectively are simulated andplotted using the kinetic parameter values given in Table 6[4 27]

The simulated concentration profiles are presented in Fig-ures 2ndash5 First in Figure 2 the time evolution of the biomassconcentration is depicted As can be seen the best matchingwith the measured data (the values of measured data plottedin all figures are taken from Table 3) is given by the PSOapproach Figure 3 presents the simulated concentrationprofiles of glucose glutamine asparagine and aspartate In allcases the PSO proposed technique ensures the best estimates(in the case of asparagine the results of Gao et al [27] arecomparable with those obtained using PSO) In Figure 4the concentration profiles of lactate proline alanine andmonoclonal antibody are plotted The best matching with


0

05

1

15

2

25

Am

mon

ia (m

M)

0

005

01

015

02

025G

luta

mat

e (m

M)

0

02

04

06

08

1

0

02

04

06

08

1

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)





Viab

le C

ells

Dea

d C

ells (1

0e6

cells

mL)

(10e6

cells

mL)

Figure 5 Simulation results profiles of concentrations GLU NH3 X and Xd

the measured data is obtained with the PSO estimation tech-nique for lactate proline and alanine In the case of antibodythe best results were obtained by Baughman et al [4] FinallyFigure 5 shows the concentration evolutions of glutamateammonia viable and dead cells respectively The glutamateand ammonia concentrations are not measured the timeprofiles of these variables were obtained from the dynamicmodel simulation The viable cells and dead cells evolutionobtained via PSO estimation match very well the measureddata

Since in industrial practice themeasured data are affectedby various disturbances one explored the extent to whichnoisy measurements affects the estimated parameter valuesFor this reason aMonte Carlo simulation approachwas usedFirst normal (Gaussian) distributions were constructed forevery measured data set in Table 3 subject to the knownmean and standard deviation of each point

A set of 150 simulated measurement sets were generatedFinally using each randomized data set a new cubic interpo-lation of the data was generated for our standard conditionand the parameter estimation problems (1198751minus1198759) were solvedfor each case

Outlying solutions were identified and excluded using abasic quartile classification method The quartile values arechosen in the following manner First use the median todivide the ordered data set into two halves The median isnot included into the halves Then the lower quartile valueis the median of the lower half of the data (119902low) and theupper quartile value is the median of the upper half of thedata (119902up) Namely the lower and upper quartile (119902low 119902up)positions were found for the set of 150 objective values andthe interquartile range 119902up minus 119902low was calculated Solutionswith objective values lying outside the interval [119902low minus

15(119902119906119901minus 119902low) 119902119906119901 + 15(119902up minus 119902low)] were considered to be

outlying cases Using the remaining solutions mean valuesand associated standard deviations were calculated for eachestimated parameter

The standard deviations were then converted to per-centages of their associated mean value These means andstandard deviations are listed in Table 7

Certain parameter estimates are much more susceptibleto variability induced through perturbations in measureddata than are others It can be seen that certain parameters(120583 11989611988911987011987813

) are less sensitive to noisy measurements than arecertain others (120593lowast

7 11987011987826 11987011987828 11987011987856

)


Table 6 Kinetic parameter estimates obtained by Gao et al [27] and by Baughman et al [4] respectively

Kinetic parameters Values (Gao et al [27]) (exponential phase) Values (Gao et al [27]) (decline phase) Values (Baughman et al [4])120593lowast

1[pmol(cell h)] 0008 minus00033 885 times 10minus4

120593lowast

2[pmol(cell h)] 00191 00058 112 times 106

120593lowast

3[pmol(cell h)] 00023 minus00014 11 times 105

120593lowast

4[pmol(cell h)] 00081 00057 197 times 10minus2

120593lowast

5[pmol(cell h)] minus001 00056 495

120593lowast

6[pmol(cell h)] minus0011 00029 134 times 105

120593lowast

7[pmol(cell h)] 06429 00573 136 times 103

120593lowast


120593lowast

9[pmol(cell h)] 00731 00113 183 times 104

11987011987811

[mM] 001 001 163 times 105

11987011987812

[mM] 001 001 108 times 104

11987011987813

[mM] 001 001 144 times 104

11987011987832

[mM] 0001 0001 964 times 102

11987011987833

[mM] 0001 0001 104 times 103

11987011987834

[mM] 0001 0001 542 times 10minus2

11987011987826

[mM] 001 001 303 times 103

11987011987827

[mM] 001 001 639 times 103

11987011987828

[mM] 001 001 373 times 10minus1

11987011987829

[mM] 001 001 323 times 105

11987011987845

[mM] 0001 0001 742 times 103

11987011987856

[mM] 0001 0001 445 times 103

120583 00399 00399 322 times 10minus2

119896119889

006 006 499 times 10minus2

The proposed modelling and parameter estimationmethod can be applied to cellular processes described bythe general form (1) and it is not yet applicable to allclasses of bioprocesses To be more specific it is hard to beapplied to processes characterized by phenomena such aspropagation reactions transport processes latency and shortintercellular phases (in epidemics) and spread (propagation)of infections that is processes with large heterogeneity anddelays A typical class of such nonlinear delay biosystemsis represented by the dynamics models describing cell-to-cell spread mechanisms encountered for example in HIVinfections [56 57]

4 Conclusions

In order to develop accurate models for mammalian cell cul-ture processes and to overcome some of the specific problemsof mAb production processes such as the nonlinearity theabsence of instrumentation and the kinetics uncertainties amultistep nonlinear particle swarm optimization-based tech-nique for the estimation of experimentally unavailable kineticparameters was designed and implemented The proposedapproach was tested by using a particular dynamical modelof mammalian cell culture as a case study but is generic forthis class of bioprocesses We have established the capability

of proposed technique to identify model parameters thatprovide an accurate simulation of experimentally observedmAbproduction process behaviourTheperformed statisticalanalysis demonstrates that the proposed estimation methodis robust against normal distributed noisy measurementsThe simulations showed that the PSO parameter estimationtechnique providesmore accurate results than those reportedin previous studies

The obtained dynamical model of the mAb productionprocess is accurate and can contribute to the developmentof model-based applications which lead to high productivityand better quality products The performed simulationsrepresent one of the possibilities of model validation Theresults show that the proposed model offers good predictionsnot only of the cell culture for instance predictions ofconcentrations of energy sources such as glutamine andglucose but also of the main amino acids and products Theproposed estimation approach can be also applied to otherbioprocesses belonging to the nonlinear class considered inthe present study

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


Table 7Monte Carlo parametermeans and standard deviation overcomplete measurement set

Kinetic parameters Mean value Standard deviation120593lowast

1[pmol(cell h)] 8127 times 10minus4 44

120593lowast

2[pmol(cell h)] 2868 times 106 51

120593lowast


120593lowast


120593lowast

5[pmol(cell h)] 7873 23

120593lowast


120593lowast


120593lowast


120593lowast


11987011987811

[mM] 8936 times 105 1711987011987812

[mM] 7030 times 104 2911987011987813

[mM] 3684 times 104 511987011987832

[mM] 731 times 102 2911987011987833

[mM] 2565 times 103 1411987011987834

[mM] 0026 1311987011987826

[mM] 2295 times 104 8711987011987827

[mM] 9273 times 103 5111987011987828

[mM] 0649 11611987011987829

[mM] 6224 times 105 3911987011987845

[mM] 1814 times 103 4411987011987856

[mM] 4657 times 104 75120583 0048 2119896119889

0070 073

Acknowledgments

This work was supported by UEFISCDI Project PACBIO no7012013 (French-Romanian project) and Project ADCOS-BIO no 2112014 PN-II-PT-PCCA-2013-4-0544

References

[1] D F Slezak C Suarez G A Cecchi G Marshall and G Stolo-vitzky ldquoWhen the optimal is not the best parameter estimationin complex biological modelsrdquo PLoS ONE vol 5 no 10 ArticleID e13283 2010

[2] C Kontoravdi S P Asprey E N Pistikopoulos and A Man-talaris ldquoDevelopment of a dynamic model of monoclonal anti-body production and glycosylation for product quality moni-toringrdquo Computers amp Chemical Engineering vol 31 no 5-6 pp392ndash400 2007

[3] C Kontoravdi E N Pistikopoulos and AMantalaris ldquoSystem-atic development of predictive mathematical models for animalcell culturesrdquo Computers amp Chemical Engineering vol 34 no 8pp 1192ndash1198 2010

[4] A C Baughman X Huang S T Sharfstein and L L MartinldquoOn the dynamic modeling of mammalian cell metabolism andmAb productionrdquo Computers amp Chemical Engineering vol 34no 2 pp 210ndash222 2010

[5] C M Batista L C S Medeiros I Eger and M J Soares ldquoMAbCZP-315D9 an antirecombinant cruzipain monoclonal anti-body that specifically labels the reservosomes of Trypanosoma

cruzi epimastigotesrdquo BioMed Research International vol 2014Article ID 714749 9 pages 2014

[6] G Sautto N Mancini G Gorini M Clementi and R BurionildquoPossible future monoclonal antibody (mAb)-Based Therapyagainst arbovirus infectionsrdquo BioMed Research Internationalvol 2013 Article ID 838491 21 pages 2013

[7] F Li N Vijayasankaran A Y Shen R Kiss and A AmanullahldquoCell culture processes for monoclonal antibody productionrdquomAbs vol 2 no 5 pp 466ndash479 2010

[8] R J Pantazes and C DMaranas ldquoMAPs a database ofmodularantibody parts for predicting tertiary structures and designingaffinitymatured antibodiesrdquoBMCBioinformatics vol 14 article168 2013

[9] A K Pavlou andM J Belsey ldquoThe therapeutic antibodies mar-ket to 2008rdquo European Journal of Pharmaceutics and Biophar-maceutics vol 59 no 3 pp 389ndash396 2005

[10] S Dhir K J Morrow R R Rhinehart and T WiesnerldquoDynamic optimization of hybridoma growth in a fed-batchbioreactorrdquo Biotechnology and Bioengineering vol 67 no 2 pp197ndash205 2000

[11] C Giersch ldquoMathematical modelling of metabolismrdquo CurrentOpinion in Plant Biology vol 3 no 3 pp 249ndash253 2000

[12] F R Sidoli AMantalaris and S P Asprey ldquoModelling of mam-malian cells and cell culture processesrdquo Cytotechnology vol 44no 1-2 pp 27ndash46 2004

[13] M Roman D Selisteanu E Bobasu and D Sendrescu ldquoMod-eling of culture cells for pharmaceutical industry applicationsrdquoin Proceedings of the 17th International Conference on SystemTheory Control and Computing (ICSTCC rsquo13) pp 459ndash464Sinaia Romania October 2013

[14] D Sendrescu and E Bobasu ldquoParameter identification of bac-terial growth bioprocesses using heuristics for global optimiza-tionrdquo in Proceedings of the 17th International Conference onSystem Theory Control and Computing (ICSTCC rsquo13) pp 485ndash490 Sinaia Romania October 2013

[15] D Karnopp and R Rosenberg System Dynamics A UnifiedApproach John Wiley amp Sons New York NY USA 1974

[16] J Thoma Introduction to Bond Graphs and Their ApplicationsPergamon Press Oxford UK 1975

[17] P Gawthrop and L Smith Metamodelling Bond Graphs andDynamic Systems Prentice Hall Hemel UK 1996

[18] W Borutzky Bond Graph Methodology Development and Anal-ysis of Multidisciplinary Dynamic System Models SpringerLondon UK 2009

[19] G Dauphin-Tanguy Ed Les bond graphs Hermes SciencesParis France 2000

[20] F Couenne C Jallut B Maschke P C Breedveld and MTayakout ldquoBond graphmodelling for chemical reactorsrdquoMath-ematical and Computer Modelling of Dynamical Systems vol 12no 2-3 pp 159ndash174 2006

[21] C Heny D Simanca and M Delgado ldquoPseudo-bond graphmodel and simulation of a continuous stirred tank reactorrdquoTheJournal of the Franklin Institute vol 337 no 1 pp 21ndash42 2000

[22] J Thoma and B Ould BouamamaModelling and Simulation inThermal and Chemical Engineering A Bond Graph ApproachSpringer Berlin Germany 2000

[23] V Dıaz-Zuccarini D Rafirou J LeFevre D R Hose and P VLawford ldquoSystemic modelling and computational physiologythe application of Bond Graph boundary conditions for 3D car-diovascular modelsrdquo Simulation Modelling Practice andTheoryvol 17 no 1 pp 125ndash136 2009


[24] D Selisteanu M Roman and D Sendrescu ldquoPseudo bondgraph modelling and on-line estimation of unknown kineticsfor a wastewater biodegradation processrdquo Simulation ModellingPractice andTheory vol 18 no 9 pp 1297ndash1313 2010

[25] M Roman and D Selisteanu ldquoPseudo bond graph modeling ofwastewater treatment bioprocessesrdquo Simulation vol 88 no 2pp 233ndash251 2012

[26] M Roman and D Selisteanu ldquoEnzymatic synthesis of ampi-cillin nonlinear modeling kinetics estimation and adaptivecontrolrdquo Journal of Biomedicine and Biotechnology vol 2012Article ID 512691 14 pages 2012

[27] J Gao V M Gorenflo J M Scharer and H M BudmanldquoDynamic metabolic modeling for a MAb bioprocessrdquo Biotech-nology Progress vol 23 no 1 pp 168ndash181 2007

[28] I Petric and V Selimbasic ldquoDevelopment and validation ofmathematical model for aerobic composting processrdquoChemicalEngineering Journal vol 139 no 2 pp 304ndash317 2008

[29] A Wachter and L T Biegler ldquoOn the implementation of aninterior-point filter line-search algorithm for large-scale non-linear programmingrdquo Mathematical Programming A Publica-tion of the Mathematical Programming Society vol 106 no 1pp 25ndash57 2006

[30] M Schwaab E C Biscaia Jr J L Monteiro and J C PintoldquoNonlinear parameter estimation through particle swarm opti-mizationrdquoChemical Engineering Science vol 63 no 6 pp 1542ndash1552 2008

[31] M C A F Rezende C B B Costa A C CostaM RWMacieland RM Filho ldquoOptimization of a large scale industrial reactorby genetic algorithmsrdquo Chemical Engineering Science vol 63no 2 pp 330ndash341 2008

[32] I A Maraziotis A Dragomir and DThanos ldquoGene regulatorynetworks modelling using a dynamic evolutionary hybridrdquoBMC Bioinformatics vol 11 article 140 2010

[33] A SırbuH J Ruskin andMCrane ldquoComparison of evolution-ary algorithms in gene regulatory network model inferencerdquoBMC Bioinformatics vol 11 article 59 2010

[34] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[35] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[36] D Sendrescu ldquoParameter identification of anaerobic wastewa-ter treatment bioprocesses using particle swarm optimizationrdquoMathematical Problems in Engineering vol 2013 Article ID103748 8 pages 2013

[37] B Shen C Liu J Ye E Feng and Z Xiu ldquoParameter iden-tification and optimization algorithm in microbial continuousculturerdquoAppliedMathematicalModelling vol 36 no 2 pp 585ndash595 2012

[38] J Zhai J Ye L Wang E Feng H Yin and Z Xiu ldquoPath-way identification using parallel optimization for a complexmetabolic system in microbial continuous culturerdquo NonlinearAnalysis Real World Applications vol 12 no 5 pp 2730ndash27412011

[39] J Yuan X Zhang X Zhu E Feng H Yin and Z XiuldquoModelling and pathway identification involving the transportmechanism of a complex metabolic system in batch culturerdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 19 no 6 pp 2088ndash2103 2014

[40] J Ye Y Zhang E Feng Z Xiu and H Yin ldquoNonlinear hybridsystem and parameter identification of microbial fed-batchculture with open loop glycerol input and pH logic controlrdquoApplied Mathematical Modelling vol 36 no 1 pp 357ndash3692012

[41] X Li S Zhang Z Xiu and E Feng ldquoParameter identificationmodel with the control term in batch anaerobic fermentationrdquoApplied Mechanics and Materials vol 217-219 pp 1535ndash15402012

[42] Z Liao and C Mei ldquoEstimation of biochemical variables usingquantumbehaved particle swarm optimization (QPSO)-trainedradius basis function neural network a case study of fermenta-tion process of L-glutamic acidrdquo African Journal of Biotechnol-ogy vol 10 no 26 pp 5203ndash5208 2011

[43] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995

[44] A Gambhir R Korke J Lee P C Fu A Europa andW S HuldquoAnalysis of cellular metabolism of hybridoma cells at distinctphysiological statesrdquo Journal of Bioscience and Bioengineeringvol 95 no 4 pp 317ndash327 2003

[45] H P J Bonarius V Hatzimanikatis K P H Meesters C D deGooijer G Schmid and J Tramper ldquoMetabolic flux analysis ofhybridoma cells in different culturemedia usingmass balancesrdquoBiotechnology and Bioengineering vol 50 no 3 pp 299ndash3181996

[46] F D Follstad R R Balcarcel G Stephanopoulos and D I CWang ldquoMetabolic flux analysis of hybridoma continuous cul-ture steady statemultiplicityrdquo Biotechnology and Bioengineeringvol 63 no 6 pp 675ndash683 1999

[47] K K Frame and W-S Hu ldquoKinetic study of hybridoma cellgrowth in continuous culture I A model for non-producingcellsrdquo Biotechnology and Bioengineering vol 37 no 1 pp 55ndash641991

[48] A Provost G Bastin S N Agathos and Y J SchneiderldquoMetabolic design of macroscopic bioreaction models applica-tion to Chinese hamster ovary cellsrdquo Bioprocess and BiosystemsEngineering vol 29 no 5-6 pp 349ndash366 2006

[49] G Bastin and D Dochain On-Line Estimation and AdaptiveControl of Bioreactors Elsevier Amsterdam The Netherlands1990

[50] D Dochain Ed Automatic Control of Bioprocesses ISTE andJohn Wiley amp Sons London UK 2008

[51] H Hasanvand B B Zad B Mozafari and H Maskani ldquoFuzzylogic controller design based SVC for improving power systemdampingrdquo International Review of Automatic Control vol 4 no5 pp 740ndash748 2011

[52] Y B Amlashi and H Afrakhte ldquoDetermination of wind plantoutput capacity using discreteMarkov chains and PSOmethodsin comparison with FCMrdquo International Review onModelling ampSimulations vol 4 no 2 pp 819ndash823 2011

[53] R C Eberhart and Y Shi ldquoComparing inertia weights and con-striction factors in particle swarm optimizationrdquo in Proceedingsof the Congress on Evolutionary Computation pp 84ndash88 LaJolla Calif USA July 2000

[54] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation pp 1945ndash1950 Washington DC USA July 1999

[55] I V Tetko D J Livingstone and A I Luik ldquoNeural networkStudies 1 Comparison of overfitting and overtrainingrdquo Journalof Chemical Information and Computer Sciences vol 35 no 5pp 826ndash833 1995


[56] R V Culshaw S Ruan and GWebb ldquoAmathematical model ofcell-to-cell spread of HIV-1 that includes a time delayrdquo Journalof Mathematical Biology vol 46 no 5 pp 425ndash444 2003

[57] S-I Niculescu C-I Morarescu W Michiels and K Gu ldquoGeo-metric ideas in the stability analysis of delay models in bio-sciencesrdquo in Biology and Control Theory Current ChallengesI Queinnec S Tarbouriech G Garcia et al Eds vol 357 ofLecture Notes in Control and Information Sciences pp 217ndash259Springer Berlin Germany 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Anatomy Research International

PeptidesInternational Journal of


Hindawi Publishing Corporation httpwwwhindawicom

International Journal of

Volume 2014

Zoology


Molecular Biology International

GenomicsInternational Journal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


BioinformaticsAdvances in

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Signal TransductionJournal of


BioMed Research International

Evolutionary BiologyInternational Journal of



Biochemistry Research International

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Genetics Research International


Advances in

Virolog y

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2014

Stem CellsInternational



Enzyme Research



Microbiology


or depleted at a rate considerably faster than the overall cellgrowth rate Consequently the concentration of each systemmetabolite and the rate of each metabolic reaction are allconsidered time-invariant [4] This approach is simple andthe obtained models are linear systems which can be easilycomputed regardless of the model size (complexity) Theinformation gathered in such pseudo-steady-state modelsconcerns the metabolic configuration of cell culture How-ever mammalian cells have a complicated internal structurewith several interconnected biochemical processes and withphenomena onmultiple time scalesThus the pseudo-steady-state models cannot describe in detail the changes that occurover a continuous time-horizon (intracellular concentrationprofiles changes in reaction rate due to gene regulation etc)Therefore the dynamic modelling is more appropriate forthese complex (and dynamical) processes In this case asystem of differential equations will describe the bioprocessmodel Inmany cases the difficulty that arises is related to thecomputational problems especially for large and stiff systemsNo matter what modelling method is chosen the complexitytogether with the nonlinearity of these processes is a limitingfactor in model building

In this paper which is an extended work of [13 14] anessential problem in dynamic modelling of cell culture sys-tems is analysed the so-called parameter estimation Themodel of such bioprocesses can be obtained by using dynamicclassical modelling (based on mass balance) or alternativeapproaches such as pseudo-bond-graph method (a versionof bond graph method introduced by Paynter in 1961 andfurther developed in [15ndash26]) However regardless of themodelling method in order to obtain a dynamical modeluseful for process development (including the design ofsome control strategies) the nonmeasurable parameters ofthe mammalian cell culture system must to be estimatedHowever any parameter in a cell culturemodel could [4] havephysical meaning and be measurable by experiment haveclear physical meaning but be experimentally inaccessibleor have no clear physical meaning (eg be purely mathe-matical in nature) Typically optimization-based techniquesare used for the estimation of nonmeasurable parametersof such biological processes [4 27 28] For example aquadratic programming technique was used by Gao et al[27] and a simple discretization scheme combined witha filtered interior point primal-dual line-search algorithm[29] was proposed by Baughman et al [4] Other nonlinearoptimization-based techniques are the genetic algorithmsorthogonal collocation and particle swarm optimizationPSO which have been applied mostly on chemical processes(see eg [30 31]) or in gene regulatory networks modelling[32 33] In order to obtain accurate solutions in the caseof the mAb production process in this paper a particleswarm-based multistep nonlinear optimization algorithm isproposed [34ndash36]

Concerning the applications of PSO for identification ofbiological systems some results were reported for the processof glycerol fermentation by Klebsiella pneumoniae in batchfed-batch and continuous cultures [37ndash41] The estimationapproach used in these works is in most cases a parallelPSO technique which requires a considerable computational

effort Another trend is related to an indirect use of PSOtechnique for estimation more precisely for the training ofa neural network which models the bioprocess [42]

During the last decade PSO algorithms have gainedmuch attention and wide applications in different fields dueto their effectiveness in performing difficult optimizationissues as well as simplicity of implementation and abilityof fast converge to a reasonably good solution PSO is apopulation-based heuristic global optimization techniquefirst introduced by Kennedy and Eberhart [43] and referredto as a swarm-intelligence technique It is motivated fromthe simulation of social behaviour of animals such as birdflocking fish schooling and swarm In this algorithm thepopulation is called a swarm and the trajectory of each parti-cle in the search space is controlled through the medium of aterm called ldquovelocityrdquo according to its own flying experienceand swarm experience in the search space

This paper proposes a multistep PSO version that usestime-varying acceleration coefficients [35] which is devel-oped to solve the nonconvex optimization problem ensuringfast convergence and very good performance Finally theobtained solution is an optimal set of the kinetic parametervalues

The proposed nonlinear modelling and estimationapproaches are analyzed in this work by using a particularmodel of mammalian cell culture as a case study but they aregeneric for this class of bioprocesses A previously publisheddynamic model of mammalian cell culture by Gao et al in[27] is used as a case study More precisely a process of anImmunoglobulin G-secreting murine hybridoma cultured ina growth medium supplemented with proline L-asparagineand L-aspartic acid is taken into consideration

2 Methods

21 MAb Synthesis by Mammalian Cell Culture ProcessDescription and Modelling Issues In order to model themammalian cell culture processes first it is necessary to ana-lyze the reconstruction of metabolic activities However thereconstruction generally includes only a subset of the highlyactivemetabolic units found in proliferatingmammalian cells[4 34] After the choice of this key subset the next stepconsists in the modelling of the reactionsrsquo rates in the frameof reconstruction This process is a very difficult one andthe modelling of reactions as single-enzyme processes byusing in vitro kinetic parameters is possible only for simpleand small reconstructions Often in vitro kinetic parametersdo not compare well against in vivo observations [4] Thereconstruction in complex processes will frequently combinea number of discrete processes into a single lumped processand will then apply kinetic parameters to the lumped processConsequently some kinetic parameters that appear in themodel may have small or no physical significance and usuallytheir values are not experimentally measurable [4]

Therefore if all of the interaction of metabolites andcell physiology are included in the modelling process thenthe size of the obtained model is very large and it is notappropriate for model-based optimization and control pur-poses The usual solution is to select a priori the elementary









119889120585

119889119905

= 119870 sdot 120593 (120585) + 119863 sdot 120585 + 119865 minus 119876 (1)

where 120585 = [1205851

1205852



[1205931

1205932



119894119895] 119894 = 1 119899 119895 = 1119898 where

119870119894119895= (plusmn)119896



119894119895is given by



11198652


supply and 119876 = [1198761

1198762





119889120585

119889119905

= 119870 sdot 120593 (120585) (2)




120585 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1205851

1205852

1205853

1205854

1205855

1205856

1205857

1205858

1205859

12058510

12058511

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198781

1198782

1198783

1198784

1198785

1198751

1198752

1198753

1198754

1198755

1198756

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(3)








2gaseous





1198891198781

119889119905


1198891198782

119889119905


1198891198783

119889119905


1198891198784

119889119905


1198891198785

119889119905


1198891198751

119889119905

= 119896611205931+ 119896621205932+ 119896631205933

1198891198752

119889119905

= 119896721205932minus 119896771205937minus 119896781205938

1198891198753

119889119905

= 1205934minus 119896871205937minus 119896881205938

1198891198754

119889119905

= 1205938

1198891198755

119889119905

= 1205937

1198891198756

119889119905

= 1205935+ 1205939

(4)


119889

119889119905

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198781

1198782

1198783

1198784

1198785

1198751

1198752

1198753

1198754

1198755

1198756

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟

120585

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[


0 0

0 0 0 0 0 minus11989626

minus11989627

minus11989628

minus1

0 minus11989632

minus11989633


minus11989638

1

0 0 0 0 minus1 1 minus11989647

minus11989648

0

0 0 11989653


minus11989658

0

11989661

11989662

11989663

0 0 0 0 0 0

0 11989672

0 0 0 0 minus11989677

minus11989678

0

0 0 0 1 0 0 minus11989687

minus11989688

0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 1 0 0

0 0 0 0 1 0 0 0 1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119870

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1205931

1205932

1205933

1205934

1205935

1205936

1205937

1205938

1205939

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟

120593(120585)

(5)



= 00508 11989626

= 1 11989627

= 00577 11989628

= 00104 11989632

= 211989633

= 2 11989637

= 00016 11989638

= 00107 11989647

= 0006 11989648

=

0072 11989653

= 2 11989657

= 00201 11989658

= 0082 11989661

= 2 11989662

= 211989663

= 2 11989672

= 2 11989677

= 00133 11989678

= 0011 11989687

= 0081and 11989688



119889119883

119889119905

= 120583119883 minus 119896119889119883119883119889 for 119905 lt 119905exp

119889119883

119889119905

= minus 119896119889119883119883119889 for 119905 ge 119905exp

119889119883119889

119889119905

= 119896119889119883119883119889

(6)




120593119894= 120593

lowast


119878119895isin119878119894

119878119895

119870119878119895119894

+ 119878119895

119894 = 1 9 (7)





119878119895119894is a kinetic half-





1 1205931= 120593lowast

1119883

1198781

11987011987811

+ 1198781

2 1205932= 120593lowast

2119883

1198781

11987011987812

+ 1198781

1198783

11987011987832

+ 1198783

3 1205933= 120593lowast

3119883

1198781

11987011987813

+ 1198781

1198783

11987011987833

+ 1198783

4 1205934= 120593lowast

4119883

1198783

11987011987834

+ 1198783

5 1205935= 120593lowast

5119883

1198784

11987011987845

+ 1198784

6 1205936= 120593lowast

6119883

1198782

11987011987826

+ 1198782

1198785

11987011987856

+ 1198785

7 1205937= 120593lowast

7119883

1198782

11987011987827

+ 1198782

8 1205938= 120593lowast

8119883

1198782

11987011987828

+ 1198782

9 1205939= 120593lowast

9119883

1198782

1198701198789+ 1198782






119889120585

119889119905

= 119870 sdot 120593 (120585) = 119891 (120585 119905 120579) (8)






119889120585 (119905)

119889119905

= 119891 (120585 119905

120579)

(9)



119882 =

1

119873 +119872

119872

sum

119895=1

119873

sum

119896=1

(120585

119896

119895minus120585

119896

119895)

2

(10)


119895and

120585119896






119894= (119909

1198941 1199091198942 119909

119894119872) and V

119894=

(V1198941 V1198942 V





119892= (119901

1198921 1199011198922 119901

119892119872) called 119892119887119890119904119905



119894


V119896+1119894

= 120596 sdot V119896119894+ 11988811199031(119901119887119890119904119905

119896

119894minus 119909

119905

119894) + 11988821199032(119892119887119890119904119905

119896

119894minus 119909

119896

119894)

119909

119896+1

119894= 119909

119896

119894+ V119896+1119894

(11)



1and 119903

2are


1+ 1198882)2 minus 1 and





V119896+1119894

= ℎ sdot [V119896119894+ 11988811199031(119901119887119890119904119905

119896

119894minus 119909

119905

119894) + 11988821199032(119892119887119890119904119905

119896

119894minus 119909

119896

119894)]

119909

119896+1

119894= 119909

119896

119894+ V119896+1119894

(12)


ℎ =

2

(


2minus 4120572

10038161003816100381610038161003816)

120572 = 1198881+ 1198882gt 4 (13)




120596 = (120596119894minus 120596119891) sdot

(119896max minus 119896)

119896max+ 120596119891 (14)




119891= 04


1is



1198881= (1198881119891minus 1198881119894) sdot

(119896max minus 119896)

119896max+ 1198881119894

1198882= (1198882119891minus 1198882119894) sdot

(119896max minus 119896)

119896max+ 1198882119894

(15)

where 1198881119894

and 1198882119894


1and 1198882and 1198881119891

and 1198882119891



1198881119894= 25 119888

2119894= 05 119888

1119891= 05 and 119888

2119891= 25 [14 35 36]









End

Yes No




Start





No

No

k = 1


k = k + 1

pbest

pbestas new pbest


Yes

Yesk le 9



1 1199052 119905

119899 to obtain a


119896 119905119896+1



119889120585 (119905119896)

119889119905

asymp

120585 (119905119896) minus 120585 (119905

119896+ 119879119904)

119879119904

(16)






lowast

7 11987011987827

P2 120593lowast

8 11987011987828

P3 120593lowast

4 11987011987834

P4 120593lowast

2 11987011987812 11987011987832

P5 120593lowast

6 11987011987826 11987011987829 120593lowast

9 11987011987856

P6 120593lowast

5 11987011987845

P7 120593lowast

3 11987011987813 11987011987833

P8 120593lowast

1 11987011987811

P9 120583 119896119889


7and119870



1198821198751=

119873

sum

119896=1

(12058510(119896 sdot 119879119904) minus

12058510(119896 sdot 119879119904))

2





120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


11987011987811

[mM] 8989 times 105

11987011987812

[mM] 6495 times 104

11987011987813

[mM] 3723 times 104

11987011987832

[mM] 7076 times 102

11987011987833

[mM] 2782 times 103

11987011987834

[mM] 001911987011987826

[mM] 2719 times 104

11987011987827

[mM] 9324 times 103

11987011987828

[mM] 053711987011987829

[mM] 6683 times 105

11987011987845

[mM] 1920 times 103

11987011987856

[mM] 4488 times 104

120583 0043119896119889

0067

st 12058510((119896 + 1) sdot 119879

119904)

=12058510(119896 sdot 119879119904) + 119879119904sdot

120593lowast


11987011987827

+ 1205852(119896 sdot 119879119904)

(17)



7and119870





0 50 100 15002468

10121416182022

Time (h)

Biom

ass (

mM

)






119894= 120593lowast





0 50 100 15015

2

25

3

35

4

Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)

Glu

cose

(mM

)

0

05

1

15

2

25

3

35

Glu

tam

ine (

mM

)

01

02

03

04

05

06

Asp

arag

ine (

mM

)

0

005

01

015

02

025

03

Asp

arta

te (m

M)

0 50 100 150Time (h)










50 100 150

05

1

15

2

25

Time (h)

Lact

ate (

mM

)

04

06

08

1

12

14

16

18

2

Ala

nine

(mM

)

03

04

05

06

07

Prol

ine (

mM

)

0

1

2

3

4

0 50 100 150Time (h)

0

50 100 150Time (h)

050 100 150Time (h)

0

times10minus4





Ant

ibod

y (10eminus4

mM

)








0

05

1

15

2

25

Am

mon

ia (m

M)

0

005

01

015

02

025G

luta

mat

e (m

M)

0

02

04

06

08

1

0

02

04

06

08

1

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)





Viab

le C

ells

Dea

d C

ells (1

0e6

cells

mL)

(10e6

cells

mL)











7 11987011987826 11987011987828 11987011987856

)





120593lowast

2[pmol(cell h)] 00191 00058 112 times 106

120593lowast


120593lowast


120593lowast


120593lowast


120593lowast

7[pmol(cell h)] 06429 00573 136 times 103

120593lowast


120593lowast

9[pmol(cell h)] 00731 00113 183 times 104

11987011987811

[mM] 001 001 163 times 105

11987011987812

[mM] 001 001 108 times 104

11987011987813

[mM] 001 001 144 times 104

11987011987832

[mM] 0001 0001 964 times 102

11987011987833

[mM] 0001 0001 104 times 103

11987011987834

[mM] 0001 0001 542 times 10minus2

11987011987826

[mM] 001 001 303 times 103

11987011987827

[mM] 001 001 639 times 103

11987011987828

[mM] 001 001 373 times 10minus1

11987011987829

[mM] 001 001 323 times 105

11987011987845

[mM] 0001 0001 742 times 103

11987011987856

[mM] 0001 0001 445 times 103

120583 00399 00399 322 times 10minus2

119896119889

006 006 499 times 10minus2


4 Conclusions










120593lowast


120593lowast


120593lowast


120593lowast

5[pmol(cell h)] 7873 23

120593lowast


120593lowast


120593lowast


120593lowast


11987011987811

[mM] 8936 times 105 1711987011987812

[mM] 7030 times 104 2911987011987813

[mM] 3684 times 104 511987011987832

[mM] 731 times 102 2911987011987833

[mM] 2565 times 103 1411987011987834

[mM] 0026 1311987011987826

[mM] 2295 times 104 8711987011987827

[mM] 9273 times 103 5111987011987828

[mM] 0649 11611987011987829

[mM] 6224 times 105 3911987011987845

[mM] 1814 times 103 4411987011987856

[mM] 4657 times 104 75120583 0048 2119896119889

0070 073

Acknowledgments


References




































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology









119889120585

119889119905

= 119870 sdot 120593 (120585) + 119863 sdot 120585 + 119865 minus 119876 (1)

where 120585 = [1205851

1205852



[1205931

1205932



119894119895] 119894 = 1 119899 119895 = 1119898 where

119870119894119895= (plusmn)119896



119894119895is given by



11198652


supply and 119876 = [1198761

1198762





119889120585

119889119905

= 119870 sdot 120593 (120585) (2)




120585 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1205851

1205852

1205853

1205854

1205855

1205856

1205857

1205858

1205859

12058510

12058511

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198781

1198782

1198783

1198784

1198785

1198751

1198752

1198753

1198754

1198755

1198756

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(3)








2gaseous





1198891198781

119889119905


1198891198782

119889119905


1198891198783

119889119905


1198891198784

119889119905


1198891198785

119889119905


1198891198751

119889119905

= 119896611205931+ 119896621205932+ 119896631205933

1198891198752

119889119905

= 119896721205932minus 119896771205937minus 119896781205938

1198891198753

119889119905

= 1205934minus 119896871205937minus 119896881205938

1198891198754

119889119905

= 1205938

1198891198755

119889119905

= 1205937

1198891198756

119889119905

= 1205935+ 1205939

(4)


119889

119889119905

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198781

1198782

1198783

1198784

1198785

1198751

1198752

1198753

1198754

1198755

1198756

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟

120585

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[


0 0

0 0 0 0 0 minus11989626

minus11989627

minus11989628

minus1

0 minus11989632

minus11989633


minus11989638

1

0 0 0 0 minus1 1 minus11989647

minus11989648

0

0 0 11989653


minus11989658

0

11989661

11989662

11989663

0 0 0 0 0 0

0 11989672

0 0 0 0 minus11989677

minus11989678

0

0 0 0 1 0 0 minus11989687

minus11989688

0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 1 0 0

0 0 0 0 1 0 0 0 1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119870

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1205931

1205932

1205933

1205934

1205935

1205936

1205937

1205938

1205939

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟

120593(120585)

(5)



= 00508 11989626

= 1 11989627

= 00577 11989628

= 00104 11989632

= 211989633

= 2 11989637

= 00016 11989638

= 00107 11989647

= 0006 11989648

=

0072 11989653

= 2 11989657

= 00201 11989658

= 0082 11989661

= 2 11989662

= 211989663

= 2 11989672

= 2 11989677

= 00133 11989678

= 0011 11989687

= 0081and 11989688



119889119883

119889119905

= 120583119883 minus 119896119889119883119883119889 for 119905 lt 119905exp

119889119883

119889119905

= minus 119896119889119883119883119889 for 119905 ge 119905exp

119889119883119889

119889119905

= 119896119889119883119883119889

(6)




120593119894= 120593

lowast


119878119895isin119878119894

119878119895

119870119878119895119894

+ 119878119895

119894 = 1 9 (7)





119878119895119894is a kinetic half-





1 1205931= 120593lowast

1119883

1198781

11987011987811

+ 1198781

2 1205932= 120593lowast

2119883

1198781

11987011987812

+ 1198781

1198783

11987011987832

+ 1198783

3 1205933= 120593lowast

3119883

1198781

11987011987813

+ 1198781

1198783

11987011987833

+ 1198783

4 1205934= 120593lowast

4119883

1198783

11987011987834

+ 1198783

5 1205935= 120593lowast

5119883

1198784

11987011987845

+ 1198784

6 1205936= 120593lowast

6119883

1198782

11987011987826

+ 1198782

1198785

11987011987856

+ 1198785

7 1205937= 120593lowast

7119883

1198782

11987011987827

+ 1198782

8 1205938= 120593lowast

8119883

1198782

11987011987828

+ 1198782

9 1205939= 120593lowast

9119883

1198782

1198701198789+ 1198782






119889120585

119889119905

= 119870 sdot 120593 (120585) = 119891 (120585 119905 120579) (8)






119889120585 (119905)

119889119905

= 119891 (120585 119905

120579)

(9)



119882 =

1

119873 +119872

119872

sum

119895=1

119873

sum

119896=1

(120585

119896

119895minus120585

119896

119895)

2

(10)


119895and

120585119896






119894= (119909

1198941 1199091198942 119909

119894119872) and V

119894=

(V1198941 V1198942 V





119892= (119901

1198921 1199011198922 119901

119892119872) called 119892119887119890119904119905



119894


V119896+1119894

= 120596 sdot V119896119894+ 11988811199031(119901119887119890119904119905

119896

119894minus 119909

119905

119894) + 11988821199032(119892119887119890119904119905

119896

119894minus 119909

119896

119894)

119909

119896+1

119894= 119909

119896

119894+ V119896+1119894

(11)



1and 119903

2are


1+ 1198882)2 minus 1 and





V119896+1119894

= ℎ sdot [V119896119894+ 11988811199031(119901119887119890119904119905

119896

119894minus 119909

119905

119894) + 11988821199032(119892119887119890119904119905

119896

119894minus 119909

119896

119894)]

119909

119896+1

119894= 119909

119896

119894+ V119896+1119894

(12)


ℎ =

2

(


2minus 4120572

10038161003816100381610038161003816)

120572 = 1198881+ 1198882gt 4 (13)




120596 = (120596119894minus 120596119891) sdot

(119896max minus 119896)

119896max+ 120596119891 (14)




119891= 04


1is



1198881= (1198881119891minus 1198881119894) sdot

(119896max minus 119896)

119896max+ 1198881119894

1198882= (1198882119891minus 1198882119894) sdot

(119896max minus 119896)

119896max+ 1198882119894

(15)

where 1198881119894

and 1198882119894


1and 1198882and 1198881119891

and 1198882119891



1198881119894= 25 119888

2119894= 05 119888

1119891= 05 and 119888

2119891= 25 [14 35 36]









End

Yes No




Start





No

No

k = 1


k = k + 1

pbest

pbestas new pbest


Yes

Yesk le 9



1 1199052 119905

119899 to obtain a


119896 119905119896+1



119889120585 (119905119896)

119889119905

asymp

120585 (119905119896) minus 120585 (119905

119896+ 119879119904)

119879119904

(16)






lowast

7 11987011987827

P2 120593lowast

8 11987011987828

P3 120593lowast

4 11987011987834

P4 120593lowast

2 11987011987812 11987011987832

P5 120593lowast

6 11987011987826 11987011987829 120593lowast

9 11987011987856

P6 120593lowast

5 11987011987845

P7 120593lowast

3 11987011987813 11987011987833

P8 120593lowast

1 11987011987811

P9 120583 119896119889


7and119870



1198821198751=

119873

sum

119896=1

(12058510(119896 sdot 119879119904) minus

12058510(119896 sdot 119879119904))

2





120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


11987011987811

[mM] 8989 times 105

11987011987812

[mM] 6495 times 104

11987011987813

[mM] 3723 times 104

11987011987832

[mM] 7076 times 102

11987011987833

[mM] 2782 times 103

11987011987834

[mM] 001911987011987826

[mM] 2719 times 104

11987011987827

[mM] 9324 times 103

11987011987828

[mM] 053711987011987829

[mM] 6683 times 105

11987011987845

[mM] 1920 times 103

11987011987856

[mM] 4488 times 104

120583 0043119896119889

0067

st 12058510((119896 + 1) sdot 119879

119904)

=12058510(119896 sdot 119879119904) + 119879119904sdot

120593lowast


11987011987827

+ 1205852(119896 sdot 119879119904)

(17)



7and119870





0 50 100 15002468

10121416182022

Time (h)

Biom

ass (

mM

)






119894= 120593lowast





0 50 100 15015

2

25

3

35

4

Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)

Glu

cose

(mM

)

0

05

1

15

2

25

3

35

Glu

tam

ine (

mM

)

01

02

03

04

05

06

Asp

arag

ine (

mM

)

0

005

01

015

02

025

03

Asp

arta

te (m

M)

0 50 100 150Time (h)










50 100 150

05

1

15

2

25

Time (h)

Lact

ate (

mM

)

04

06

08

1

12

14

16

18

2

Ala

nine

(mM

)

03

04

05

06

07

Prol

ine (

mM

)

0

1

2

3

4

0 50 100 150Time (h)

0

50 100 150Time (h)

050 100 150Time (h)

0

times10minus4





Ant

ibod

y (10eminus4

mM

)








0

05

1

15

2

25

Am

mon

ia (m

M)

0

005

01

015

02

025G

luta

mat

e (m

M)

0

02

04

06

08

1

0

02

04

06

08

1

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)





Viab

le C

ells

Dea

d C

ells (1

0e6

cells

mL)

(10e6

cells

mL)











7 11987011987826 11987011987828 11987011987856

)





120593lowast

2[pmol(cell h)] 00191 00058 112 times 106

120593lowast


120593lowast


120593lowast


120593lowast


120593lowast

7[pmol(cell h)] 06429 00573 136 times 103

120593lowast


120593lowast

9[pmol(cell h)] 00731 00113 183 times 104

11987011987811

[mM] 001 001 163 times 105

11987011987812

[mM] 001 001 108 times 104

11987011987813

[mM] 001 001 144 times 104

11987011987832

[mM] 0001 0001 964 times 102

11987011987833

[mM] 0001 0001 104 times 103

11987011987834

[mM] 0001 0001 542 times 10minus2

11987011987826

[mM] 001 001 303 times 103

11987011987827

[mM] 001 001 639 times 103

11987011987828

[mM] 001 001 373 times 10minus1

11987011987829

[mM] 001 001 323 times 105

11987011987845

[mM] 0001 0001 742 times 103

11987011987856

[mM] 0001 0001 445 times 103

120583 00399 00399 322 times 10minus2

119896119889

006 006 499 times 10minus2


4 Conclusions










120593lowast


120593lowast


120593lowast


120593lowast

5[pmol(cell h)] 7873 23

120593lowast


120593lowast


120593lowast


120593lowast


11987011987811

[mM] 8936 times 105 1711987011987812

[mM] 7030 times 104 2911987011987813

[mM] 3684 times 104 511987011987832

[mM] 731 times 102 2911987011987833

[mM] 2565 times 103 1411987011987834

[mM] 0026 1311987011987826

[mM] 2295 times 104 8711987011987827

[mM] 9273 times 103 5111987011987828

[mM] 0649 11611987011987829

[mM] 6224 times 105 3911987011987845

[mM] 1814 times 103 4411987011987856

[mM] 4657 times 104 75120583 0048 2119896119889

0070 073

Acknowledgments


References




































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology



120585 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1205851

1205852

1205853

1205854

1205855

1205856

1205857

1205858

1205859

12058510

12058511

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198781

1198782

1198783

1198784

1198785

1198751

1198752

1198753

1198754

1198755

1198756

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(3)








2gaseous





1198891198781

119889119905


1198891198782

119889119905


1198891198783

119889119905


1198891198784

119889119905


1198891198785

119889119905


1198891198751

119889119905

= 119896611205931+ 119896621205932+ 119896631205933

1198891198752

119889119905

= 119896721205932minus 119896771205937minus 119896781205938

1198891198753

119889119905

= 1205934minus 119896871205937minus 119896881205938

1198891198754

119889119905

= 1205938

1198891198755

119889119905

= 1205937

1198891198756

119889119905

= 1205935+ 1205939

(4)


119889

119889119905

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1198781

1198782

1198783

1198784

1198785

1198751

1198752

1198753

1198754

1198755

1198756

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟

120585

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[


0 0

0 0 0 0 0 minus11989626

minus11989627

minus11989628

minus1

0 minus11989632

minus11989633


minus11989638

1

0 0 0 0 minus1 1 minus11989647

minus11989648

0

0 0 11989653


minus11989658

0

11989661

11989662

11989663

0 0 0 0 0 0

0 11989672

0 0 0 0 minus11989677

minus11989678

0

0 0 0 1 0 0 minus11989687

minus11989688

0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 1 0 0

0 0 0 0 1 0 0 0 1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119870

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1205931

1205932

1205933

1205934

1205935

1205936

1205937

1205938

1205939

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]⏟⏟⏟⏟⏟⏟⏟⏟⏟

120593(120585)

(5)



= 00508 11989626

= 1 11989627

= 00577 11989628

= 00104 11989632

= 211989633

= 2 11989637

= 00016 11989638

= 00107 11989647

= 0006 11989648

=

0072 11989653

= 2 11989657

= 00201 11989658

= 0082 11989661

= 2 11989662

= 211989663

= 2 11989672

= 2 11989677

= 00133 11989678

= 0011 11989687

= 0081and 11989688



119889119883

119889119905

= 120583119883 minus 119896119889119883119883119889 for 119905 lt 119905exp

119889119883

119889119905

= minus 119896119889119883119883119889 for 119905 ge 119905exp

119889119883119889

119889119905

= 119896119889119883119883119889

(6)




120593119894= 120593

lowast


119878119895isin119878119894

119878119895

119870119878119895119894

+ 119878119895

119894 = 1 9 (7)





119878119895119894is a kinetic half-





1 1205931= 120593lowast

1119883

1198781

11987011987811

+ 1198781

2 1205932= 120593lowast

2119883

1198781

11987011987812

+ 1198781

1198783

11987011987832

+ 1198783

3 1205933= 120593lowast

3119883

1198781

11987011987813

+ 1198781

1198783

11987011987833

+ 1198783

4 1205934= 120593lowast

4119883

1198783

11987011987834

+ 1198783

5 1205935= 120593lowast

5119883

1198784

11987011987845

+ 1198784

6 1205936= 120593lowast

6119883

1198782

11987011987826

+ 1198782

1198785

11987011987856

+ 1198785

7 1205937= 120593lowast

7119883

1198782

11987011987827

+ 1198782

8 1205938= 120593lowast

8119883

1198782

11987011987828

+ 1198782

9 1205939= 120593lowast

9119883

1198782

1198701198789+ 1198782






119889120585

119889119905

= 119870 sdot 120593 (120585) = 119891 (120585 119905 120579) (8)






119889120585 (119905)

119889119905

= 119891 (120585 119905

120579)

(9)



119882 =

1

119873 +119872

119872

sum

119895=1

119873

sum

119896=1

(120585

119896

119895minus120585

119896

119895)

2

(10)


119895and

120585119896






119894= (119909

1198941 1199091198942 119909

119894119872) and V

119894=

(V1198941 V1198942 V





119892= (119901

1198921 1199011198922 119901

119892119872) called 119892119887119890119904119905



119894


V119896+1119894

= 120596 sdot V119896119894+ 11988811199031(119901119887119890119904119905

119896

119894minus 119909

119905

119894) + 11988821199032(119892119887119890119904119905

119896

119894minus 119909

119896

119894)

119909

119896+1

119894= 119909

119896

119894+ V119896+1119894

(11)



1and 119903

2are


1+ 1198882)2 minus 1 and





V119896+1119894

= ℎ sdot [V119896119894+ 11988811199031(119901119887119890119904119905

119896

119894minus 119909

119905

119894) + 11988821199032(119892119887119890119904119905

119896

119894minus 119909

119896

119894)]

119909

119896+1

119894= 119909

119896

119894+ V119896+1119894

(12)


ℎ =

2

(


2minus 4120572

10038161003816100381610038161003816)

120572 = 1198881+ 1198882gt 4 (13)




120596 = (120596119894minus 120596119891) sdot

(119896max minus 119896)

119896max+ 120596119891 (14)




119891= 04


1is



1198881= (1198881119891minus 1198881119894) sdot

(119896max minus 119896)

119896max+ 1198881119894

1198882= (1198882119891minus 1198882119894) sdot

(119896max minus 119896)

119896max+ 1198882119894

(15)

where 1198881119894

and 1198882119894


1and 1198882and 1198881119891

and 1198882119891



1198881119894= 25 119888

2119894= 05 119888

1119891= 05 and 119888

2119891= 25 [14 35 36]









End

Yes No




Start





No

No

k = 1


k = k + 1

pbest

pbestas new pbest


Yes

Yesk le 9



1 1199052 119905

119899 to obtain a


119896 119905119896+1



119889120585 (119905119896)

119889119905

asymp

120585 (119905119896) minus 120585 (119905

119896+ 119879119904)

119879119904

(16)






lowast

7 11987011987827

P2 120593lowast

8 11987011987828

P3 120593lowast

4 11987011987834

P4 120593lowast

2 11987011987812 11987011987832

P5 120593lowast

6 11987011987826 11987011987829 120593lowast

9 11987011987856

P6 120593lowast

5 11987011987845

P7 120593lowast

3 11987011987813 11987011987833

P8 120593lowast

1 11987011987811

P9 120583 119896119889


7and119870



1198821198751=

119873

sum

119896=1

(12058510(119896 sdot 119879119904) minus

12058510(119896 sdot 119879119904))

2





120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


11987011987811

[mM] 8989 times 105

11987011987812

[mM] 6495 times 104

11987011987813

[mM] 3723 times 104

11987011987832

[mM] 7076 times 102

11987011987833

[mM] 2782 times 103

11987011987834

[mM] 001911987011987826

[mM] 2719 times 104

11987011987827

[mM] 9324 times 103

11987011987828

[mM] 053711987011987829

[mM] 6683 times 105

11987011987845

[mM] 1920 times 103

11987011987856

[mM] 4488 times 104

120583 0043119896119889

0067

st 12058510((119896 + 1) sdot 119879

119904)

=12058510(119896 sdot 119879119904) + 119879119904sdot

120593lowast


11987011987827

+ 1205852(119896 sdot 119879119904)

(17)



7and119870





0 50 100 15002468

10121416182022

Time (h)

Biom

ass (

mM

)






119894= 120593lowast





0 50 100 15015

2

25

3

35

4

Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)

Glu

cose

(mM

)

0

05

1

15

2

25

3

35

Glu

tam

ine (

mM

)

01

02

03

04

05

06

Asp

arag

ine (

mM

)

0

005

01

015

02

025

03

Asp

arta

te (m

M)

0 50 100 150Time (h)










50 100 150

05

1

15

2

25

Time (h)

Lact

ate (

mM

)

04

06

08

1

12

14

16

18

2

Ala

nine

(mM

)

03

04

05

06

07

Prol

ine (

mM

)

0

1

2

3

4

0 50 100 150Time (h)

0

50 100 150Time (h)

050 100 150Time (h)

0

times10minus4





Ant

ibod

y (10eminus4

mM

)








0

05

1

15

2

25

Am

mon

ia (m

M)

0

005

01

015

02

025G

luta

mat

e (m

M)

0

02

04

06

08

1

0

02

04

06

08

1

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)





Viab

le C

ells

Dea

d C

ells (1

0e6

cells

mL)

(10e6

cells

mL)











7 11987011987826 11987011987828 11987011987856

)





120593lowast

2[pmol(cell h)] 00191 00058 112 times 106

120593lowast


120593lowast


120593lowast


120593lowast


120593lowast

7[pmol(cell h)] 06429 00573 136 times 103

120593lowast


120593lowast

9[pmol(cell h)] 00731 00113 183 times 104

11987011987811

[mM] 001 001 163 times 105

11987011987812

[mM] 001 001 108 times 104

11987011987813

[mM] 001 001 144 times 104

11987011987832

[mM] 0001 0001 964 times 102

11987011987833

[mM] 0001 0001 104 times 103

11987011987834

[mM] 0001 0001 542 times 10minus2

11987011987826

[mM] 001 001 303 times 103

11987011987827

[mM] 001 001 639 times 103

11987011987828

[mM] 001 001 373 times 10minus1

11987011987829

[mM] 001 001 323 times 105

11987011987845

[mM] 0001 0001 742 times 103

11987011987856

[mM] 0001 0001 445 times 103

120583 00399 00399 322 times 10minus2

119896119889

006 006 499 times 10minus2


4 Conclusions










120593lowast


120593lowast


120593lowast


120593lowast

5[pmol(cell h)] 7873 23

120593lowast


120593lowast


120593lowast


120593lowast


11987011987811

[mM] 8936 times 105 1711987011987812

[mM] 7030 times 104 2911987011987813

[mM] 3684 times 104 511987011987832

[mM] 731 times 102 2911987011987833

[mM] 2565 times 103 1411987011987834

[mM] 0026 1311987011987826

[mM] 2295 times 104 8711987011987827

[mM] 9273 times 103 5111987011987828

[mM] 0649 11611987011987829

[mM] 6224 times 105 3911987011987845

[mM] 1814 times 103 4411987011987856

[mM] 4657 times 104 75120583 0048 2119896119889

0070 073

Acknowledgments


References




































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology



= 00508 11989626

= 1 11989627

= 00577 11989628

= 00104 11989632

= 211989633

= 2 11989637

= 00016 11989638

= 00107 11989647

= 0006 11989648

=

0072 11989653

= 2 11989657

= 00201 11989658

= 0082 11989661

= 2 11989662

= 211989663

= 2 11989672

= 2 11989677

= 00133 11989678

= 0011 11989687

= 0081and 11989688



119889119883

119889119905

= 120583119883 minus 119896119889119883119883119889 for 119905 lt 119905exp

119889119883

119889119905

= minus 119896119889119883119883119889 for 119905 ge 119905exp

119889119883119889

119889119905

= 119896119889119883119883119889

(6)




120593119894= 120593

lowast


119878119895isin119878119894

119878119895

119870119878119895119894

+ 119878119895

119894 = 1 9 (7)





119878119895119894is a kinetic half-





1 1205931= 120593lowast

1119883

1198781

11987011987811

+ 1198781

2 1205932= 120593lowast

2119883

1198781

11987011987812

+ 1198781

1198783

11987011987832

+ 1198783

3 1205933= 120593lowast

3119883

1198781

11987011987813

+ 1198781

1198783

11987011987833

+ 1198783

4 1205934= 120593lowast

4119883

1198783

11987011987834

+ 1198783

5 1205935= 120593lowast

5119883

1198784

11987011987845

+ 1198784

6 1205936= 120593lowast

6119883

1198782

11987011987826

+ 1198782

1198785

11987011987856

+ 1198785

7 1205937= 120593lowast

7119883

1198782

11987011987827

+ 1198782

8 1205938= 120593lowast

8119883

1198782

11987011987828

+ 1198782

9 1205939= 120593lowast

9119883

1198782

1198701198789+ 1198782






119889120585

119889119905

= 119870 sdot 120593 (120585) = 119891 (120585 119905 120579) (8)






119889120585 (119905)

119889119905

= 119891 (120585 119905

120579)

(9)



119882 =

1

119873 +119872

119872

sum

119895=1

119873

sum

119896=1

(120585

119896

119895minus120585

119896

119895)

2

(10)


119895and

120585119896






119894= (119909

1198941 1199091198942 119909

119894119872) and V

119894=

(V1198941 V1198942 V





119892= (119901

1198921 1199011198922 119901

119892119872) called 119892119887119890119904119905



119894


V119896+1119894

= 120596 sdot V119896119894+ 11988811199031(119901119887119890119904119905

119896

119894minus 119909

119905

119894) + 11988821199032(119892119887119890119904119905

119896

119894minus 119909

119896

119894)

119909

119896+1

119894= 119909

119896

119894+ V119896+1119894

(11)



1and 119903

2are


1+ 1198882)2 minus 1 and





V119896+1119894

= ℎ sdot [V119896119894+ 11988811199031(119901119887119890119904119905

119896

119894minus 119909

119905

119894) + 11988821199032(119892119887119890119904119905

119896

119894minus 119909

119896

119894)]

119909

119896+1

119894= 119909

119896

119894+ V119896+1119894

(12)


ℎ =

2

(


2minus 4120572

10038161003816100381610038161003816)

120572 = 1198881+ 1198882gt 4 (13)




120596 = (120596119894minus 120596119891) sdot

(119896max minus 119896)

119896max+ 120596119891 (14)




119891= 04


1is



1198881= (1198881119891minus 1198881119894) sdot

(119896max minus 119896)

119896max+ 1198881119894

1198882= (1198882119891minus 1198882119894) sdot

(119896max minus 119896)

119896max+ 1198882119894

(15)

where 1198881119894

and 1198882119894


1and 1198882and 1198881119891

and 1198882119891



1198881119894= 25 119888

2119894= 05 119888

1119891= 05 and 119888

2119891= 25 [14 35 36]









End

Yes No




Start





No

No

k = 1


k = k + 1

pbest

pbestas new pbest


Yes

Yesk le 9



1 1199052 119905

119899 to obtain a


119896 119905119896+1



119889120585 (119905119896)

119889119905

asymp

120585 (119905119896) minus 120585 (119905

119896+ 119879119904)

119879119904

(16)






lowast

7 11987011987827

P2 120593lowast

8 11987011987828

P3 120593lowast

4 11987011987834

P4 120593lowast

2 11987011987812 11987011987832

P5 120593lowast

6 11987011987826 11987011987829 120593lowast

9 11987011987856

P6 120593lowast

5 11987011987845

P7 120593lowast

3 11987011987813 11987011987833

P8 120593lowast

1 11987011987811

P9 120583 119896119889


7and119870



1198821198751=

119873

sum

119896=1

(12058510(119896 sdot 119879119904) minus

12058510(119896 sdot 119879119904))

2





120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


11987011987811

[mM] 8989 times 105

11987011987812

[mM] 6495 times 104

11987011987813

[mM] 3723 times 104

11987011987832

[mM] 7076 times 102

11987011987833

[mM] 2782 times 103

11987011987834

[mM] 001911987011987826

[mM] 2719 times 104

11987011987827

[mM] 9324 times 103

11987011987828

[mM] 053711987011987829

[mM] 6683 times 105

11987011987845

[mM] 1920 times 103

11987011987856

[mM] 4488 times 104

120583 0043119896119889

0067

st 12058510((119896 + 1) sdot 119879

119904)

=12058510(119896 sdot 119879119904) + 119879119904sdot

120593lowast


11987011987827

+ 1205852(119896 sdot 119879119904)

(17)



7and119870





0 50 100 15002468

10121416182022

Time (h)

Biom

ass (

mM

)






119894= 120593lowast





0 50 100 15015

2

25

3

35

4

Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)

Glu

cose

(mM

)

0

05

1

15

2

25

3

35

Glu

tam

ine (

mM

)

01

02

03

04

05

06

Asp

arag

ine (

mM

)

0

005

01

015

02

025

03

Asp

arta

te (m

M)

0 50 100 150Time (h)










50 100 150

05

1

15

2

25

Time (h)

Lact

ate (

mM

)

04

06

08

1

12

14

16

18

2

Ala

nine

(mM

)

03

04

05

06

07

Prol

ine (

mM

)

0

1

2

3

4

0 50 100 150Time (h)

0

50 100 150Time (h)

050 100 150Time (h)

0

times10minus4





Ant

ibod

y (10eminus4

mM

)








0

05

1

15

2

25

Am

mon

ia (m

M)

0

005

01

015

02

025G

luta

mat

e (m

M)

0

02

04

06

08

1

0

02

04

06

08

1

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)





Viab

le C

ells

Dea

d C

ells (1

0e6

cells

mL)

(10e6

cells

mL)











7 11987011987826 11987011987828 11987011987856

)





120593lowast

2[pmol(cell h)] 00191 00058 112 times 106

120593lowast


120593lowast


120593lowast


120593lowast


120593lowast

7[pmol(cell h)] 06429 00573 136 times 103

120593lowast


120593lowast

9[pmol(cell h)] 00731 00113 183 times 104

11987011987811

[mM] 001 001 163 times 105

11987011987812

[mM] 001 001 108 times 104

11987011987813

[mM] 001 001 144 times 104

11987011987832

[mM] 0001 0001 964 times 102

11987011987833

[mM] 0001 0001 104 times 103

11987011987834

[mM] 0001 0001 542 times 10minus2

11987011987826

[mM] 001 001 303 times 103

11987011987827

[mM] 001 001 639 times 103

11987011987828

[mM] 001 001 373 times 10minus1

11987011987829

[mM] 001 001 323 times 105

11987011987845

[mM] 0001 0001 742 times 103

11987011987856

[mM] 0001 0001 445 times 103

120583 00399 00399 322 times 10minus2

119896119889

006 006 499 times 10minus2


4 Conclusions










120593lowast


120593lowast


120593lowast


120593lowast

5[pmol(cell h)] 7873 23

120593lowast


120593lowast


120593lowast


120593lowast


11987011987811

[mM] 8936 times 105 1711987011987812

[mM] 7030 times 104 2911987011987813

[mM] 3684 times 104 511987011987832

[mM] 731 times 102 2911987011987833

[mM] 2565 times 103 1411987011987834

[mM] 0026 1311987011987826

[mM] 2295 times 104 8711987011987827

[mM] 9273 times 103 5111987011987828

[mM] 0649 11611987011987829

[mM] 6224 times 105 3911987011987845

[mM] 1814 times 103 4411987011987856

[mM] 4657 times 104 75120583 0048 2119896119889

0070 073

Acknowledgments


References




































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology





119889120585 (119905)

119889119905

= 119891 (120585 119905

120579)

(9)



119882 =

1

119873 +119872

119872

sum

119895=1

119873

sum

119896=1

(120585

119896

119895minus120585

119896

119895)

2

(10)


119895and

120585119896






119894= (119909

1198941 1199091198942 119909

119894119872) and V

119894=

(V1198941 V1198942 V





119892= (119901

1198921 1199011198922 119901

119892119872) called 119892119887119890119904119905



119894


V119896+1119894

= 120596 sdot V119896119894+ 11988811199031(119901119887119890119904119905

119896

119894minus 119909

119905

119894) + 11988821199032(119892119887119890119904119905

119896

119894minus 119909

119896

119894)

119909

119896+1

119894= 119909

119896

119894+ V119896+1119894

(11)



1and 119903

2are


1+ 1198882)2 minus 1 and





V119896+1119894

= ℎ sdot [V119896119894+ 11988811199031(119901119887119890119904119905

119896

119894minus 119909

119905

119894) + 11988821199032(119892119887119890119904119905

119896

119894minus 119909

119896

119894)]

119909

119896+1

119894= 119909

119896

119894+ V119896+1119894

(12)


ℎ =

2

(


2minus 4120572

10038161003816100381610038161003816)

120572 = 1198881+ 1198882gt 4 (13)




120596 = (120596119894minus 120596119891) sdot

(119896max minus 119896)

119896max+ 120596119891 (14)




119891= 04


1is



1198881= (1198881119891minus 1198881119894) sdot

(119896max minus 119896)

119896max+ 1198881119894

1198882= (1198882119891minus 1198882119894) sdot

(119896max minus 119896)

119896max+ 1198882119894

(15)

where 1198881119894

and 1198882119894


1and 1198882and 1198881119891

and 1198882119891



1198881119894= 25 119888

2119894= 05 119888

1119891= 05 and 119888

2119891= 25 [14 35 36]









End

Yes No




Start





No

No

k = 1


k = k + 1

pbest

pbestas new pbest


Yes

Yesk le 9



1 1199052 119905

119899 to obtain a


119896 119905119896+1



119889120585 (119905119896)

119889119905

asymp

120585 (119905119896) minus 120585 (119905

119896+ 119879119904)

119879119904

(16)






lowast

7 11987011987827

P2 120593lowast

8 11987011987828

P3 120593lowast

4 11987011987834

P4 120593lowast

2 11987011987812 11987011987832

P5 120593lowast

6 11987011987826 11987011987829 120593lowast

9 11987011987856

P6 120593lowast

5 11987011987845

P7 120593lowast

3 11987011987813 11987011987833

P8 120593lowast

1 11987011987811

P9 120583 119896119889


7and119870



1198821198751=

119873

sum

119896=1

(12058510(119896 sdot 119879119904) minus

12058510(119896 sdot 119879119904))

2





120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


11987011987811

[mM] 8989 times 105

11987011987812

[mM] 6495 times 104

11987011987813

[mM] 3723 times 104

11987011987832

[mM] 7076 times 102

11987011987833

[mM] 2782 times 103

11987011987834

[mM] 001911987011987826

[mM] 2719 times 104

11987011987827

[mM] 9324 times 103

11987011987828

[mM] 053711987011987829

[mM] 6683 times 105

11987011987845

[mM] 1920 times 103

11987011987856

[mM] 4488 times 104

120583 0043119896119889

0067

st 12058510((119896 + 1) sdot 119879

119904)

=12058510(119896 sdot 119879119904) + 119879119904sdot

120593lowast


11987011987827

+ 1205852(119896 sdot 119879119904)

(17)



7and119870





0 50 100 15002468

10121416182022

Time (h)

Biom

ass (

mM

)






119894= 120593lowast





0 50 100 15015

2

25

3

35

4

Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)

Glu

cose

(mM

)

0

05

1

15

2

25

3

35

Glu

tam

ine (

mM

)

01

02

03

04

05

06

Asp

arag

ine (

mM

)

0

005

01

015

02

025

03

Asp

arta

te (m

M)

0 50 100 150Time (h)










50 100 150

05

1

15

2

25

Time (h)

Lact

ate (

mM

)

04

06

08

1

12

14

16

18

2

Ala

nine

(mM

)

03

04

05

06

07

Prol

ine (

mM

)

0

1

2

3

4

0 50 100 150Time (h)

0

50 100 150Time (h)

050 100 150Time (h)

0

times10minus4





Ant

ibod

y (10eminus4

mM

)








0

05

1

15

2

25

Am

mon

ia (m

M)

0

005

01

015

02

025G

luta

mat

e (m

M)

0

02

04

06

08

1

0

02

04

06

08

1

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)





Viab

le C

ells

Dea

d C

ells (1

0e6

cells

mL)

(10e6

cells

mL)











7 11987011987826 11987011987828 11987011987856

)





120593lowast

2[pmol(cell h)] 00191 00058 112 times 106

120593lowast


120593lowast


120593lowast


120593lowast


120593lowast

7[pmol(cell h)] 06429 00573 136 times 103

120593lowast


120593lowast

9[pmol(cell h)] 00731 00113 183 times 104

11987011987811

[mM] 001 001 163 times 105

11987011987812

[mM] 001 001 108 times 104

11987011987813

[mM] 001 001 144 times 104

11987011987832

[mM] 0001 0001 964 times 102

11987011987833

[mM] 0001 0001 104 times 103

11987011987834

[mM] 0001 0001 542 times 10minus2

11987011987826

[mM] 001 001 303 times 103

11987011987827

[mM] 001 001 639 times 103

11987011987828

[mM] 001 001 373 times 10minus1

11987011987829

[mM] 001 001 323 times 105

11987011987845

[mM] 0001 0001 742 times 103

11987011987856

[mM] 0001 0001 445 times 103

120583 00399 00399 322 times 10minus2

119896119889

006 006 499 times 10minus2


4 Conclusions










120593lowast


120593lowast


120593lowast


120593lowast

5[pmol(cell h)] 7873 23

120593lowast


120593lowast


120593lowast


120593lowast


11987011987811

[mM] 8936 times 105 1711987011987812

[mM] 7030 times 104 2911987011987813

[mM] 3684 times 104 511987011987832

[mM] 731 times 102 2911987011987833

[mM] 2565 times 103 1411987011987834

[mM] 0026 1311987011987826

[mM] 2295 times 104 8711987011987827

[mM] 9273 times 103 5111987011987828

[mM] 0649 11611987011987829

[mM] 6224 times 105 3911987011987845

[mM] 1814 times 103 4411987011987856

[mM] 4657 times 104 75120583 0048 2119896119889

0070 073

Acknowledgments


References




































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology




120596 = (120596119894minus 120596119891) sdot

(119896max minus 119896)

119896max+ 120596119891 (14)




119891= 04


1is



1198881= (1198881119891minus 1198881119894) sdot

(119896max minus 119896)

119896max+ 1198881119894

1198882= (1198882119891minus 1198882119894) sdot

(119896max minus 119896)

119896max+ 1198882119894

(15)

where 1198881119894

and 1198882119894


1and 1198882and 1198881119891

and 1198882119891



1198881119894= 25 119888

2119894= 05 119888

1119891= 05 and 119888

2119891= 25 [14 35 36]









End

Yes No




Start





No

No

k = 1


k = k + 1

pbest

pbestas new pbest


Yes

Yesk le 9



1 1199052 119905

119899 to obtain a


119896 119905119896+1



119889120585 (119905119896)

119889119905

asymp

120585 (119905119896) minus 120585 (119905

119896+ 119879119904)

119879119904

(16)






lowast

7 11987011987827

P2 120593lowast

8 11987011987828

P3 120593lowast

4 11987011987834

P4 120593lowast

2 11987011987812 11987011987832

P5 120593lowast

6 11987011987826 11987011987829 120593lowast

9 11987011987856

P6 120593lowast

5 11987011987845

P7 120593lowast

3 11987011987813 11987011987833

P8 120593lowast

1 11987011987811

P9 120583 119896119889


7and119870



1198821198751=

119873

sum

119896=1

(12058510(119896 sdot 119879119904) minus

12058510(119896 sdot 119879119904))

2





120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


11987011987811

[mM] 8989 times 105

11987011987812

[mM] 6495 times 104

11987011987813

[mM] 3723 times 104

11987011987832

[mM] 7076 times 102

11987011987833

[mM] 2782 times 103

11987011987834

[mM] 001911987011987826

[mM] 2719 times 104

11987011987827

[mM] 9324 times 103

11987011987828

[mM] 053711987011987829

[mM] 6683 times 105

11987011987845

[mM] 1920 times 103

11987011987856

[mM] 4488 times 104

120583 0043119896119889

0067

st 12058510((119896 + 1) sdot 119879

119904)

=12058510(119896 sdot 119879119904) + 119879119904sdot

120593lowast


11987011987827

+ 1205852(119896 sdot 119879119904)

(17)



7and119870





0 50 100 15002468

10121416182022

Time (h)

Biom

ass (

mM

)






119894= 120593lowast





0 50 100 15015

2

25

3

35

4

Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)

Glu

cose

(mM

)

0

05

1

15

2

25

3

35

Glu

tam

ine (

mM

)

01

02

03

04

05

06

Asp

arag

ine (

mM

)

0

005

01

015

02

025

03

Asp

arta

te (m

M)

0 50 100 150Time (h)










50 100 150

05

1

15

2

25

Time (h)

Lact

ate (

mM

)

04

06

08

1

12

14

16

18

2

Ala

nine

(mM

)

03

04

05

06

07

Prol

ine (

mM

)

0

1

2

3

4

0 50 100 150Time (h)

0

50 100 150Time (h)

050 100 150Time (h)

0

times10minus4





Ant

ibod

y (10eminus4

mM

)








0

05

1

15

2

25

Am

mon

ia (m

M)

0

005

01

015

02

025G

luta

mat

e (m

M)

0

02

04

06

08

1

0

02

04

06

08

1

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)





Viab

le C

ells

Dea

d C

ells (1

0e6

cells

mL)

(10e6

cells

mL)











7 11987011987826 11987011987828 11987011987856

)





120593lowast

2[pmol(cell h)] 00191 00058 112 times 106

120593lowast


120593lowast


120593lowast


120593lowast


120593lowast

7[pmol(cell h)] 06429 00573 136 times 103

120593lowast


120593lowast

9[pmol(cell h)] 00731 00113 183 times 104

11987011987811

[mM] 001 001 163 times 105

11987011987812

[mM] 001 001 108 times 104

11987011987813

[mM] 001 001 144 times 104

11987011987832

[mM] 0001 0001 964 times 102

11987011987833

[mM] 0001 0001 104 times 103

11987011987834

[mM] 0001 0001 542 times 10minus2

11987011987826

[mM] 001 001 303 times 103

11987011987827

[mM] 001 001 639 times 103

11987011987828

[mM] 001 001 373 times 10minus1

11987011987829

[mM] 001 001 323 times 105

11987011987845

[mM] 0001 0001 742 times 103

11987011987856

[mM] 0001 0001 445 times 103

120583 00399 00399 322 times 10minus2

119896119889

006 006 499 times 10minus2


4 Conclusions










120593lowast


120593lowast


120593lowast


120593lowast

5[pmol(cell h)] 7873 23

120593lowast


120593lowast


120593lowast


120593lowast


11987011987811

[mM] 8936 times 105 1711987011987812

[mM] 7030 times 104 2911987011987813

[mM] 3684 times 104 511987011987832

[mM] 731 times 102 2911987011987833

[mM] 2565 times 103 1411987011987834

[mM] 0026 1311987011987826

[mM] 2295 times 104 8711987011987827

[mM] 9273 times 103 5111987011987828

[mM] 0649 11611987011987829

[mM] 6224 times 105 3911987011987845

[mM] 1814 times 103 4411987011987856

[mM] 4657 times 104 75120583 0048 2119896119889

0070 073

Acknowledgments


References




































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


End

Yes No




Start





No

No

k = 1


k = k + 1

pbest

pbestas new pbest


Yes

Yesk le 9



1 1199052 119905

119899 to obtain a


119896 119905119896+1



119889120585 (119905119896)

119889119905

asymp

120585 (119905119896) minus 120585 (119905

119896+ 119879119904)

119879119904

(16)






lowast

7 11987011987827

P2 120593lowast

8 11987011987828

P3 120593lowast

4 11987011987834

P4 120593lowast

2 11987011987812 11987011987832

P5 120593lowast

6 11987011987826 11987011987829 120593lowast

9 11987011987856

P6 120593lowast

5 11987011987845

P7 120593lowast

3 11987011987813 11987011987833

P8 120593lowast

1 11987011987811

P9 120583 119896119889


7and119870



1198821198751=

119873

sum

119896=1

(12058510(119896 sdot 119879119904) minus

12058510(119896 sdot 119879119904))

2





120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


11987011987811

[mM] 8989 times 105

11987011987812

[mM] 6495 times 104

11987011987813

[mM] 3723 times 104

11987011987832

[mM] 7076 times 102

11987011987833

[mM] 2782 times 103

11987011987834

[mM] 001911987011987826

[mM] 2719 times 104

11987011987827

[mM] 9324 times 103

11987011987828

[mM] 053711987011987829

[mM] 6683 times 105

11987011987845

[mM] 1920 times 103

11987011987856

[mM] 4488 times 104

120583 0043119896119889

0067

st 12058510((119896 + 1) sdot 119879

119904)

=12058510(119896 sdot 119879119904) + 119879119904sdot

120593lowast


11987011987827

+ 1205852(119896 sdot 119879119904)

(17)



7and119870





0 50 100 15002468

10121416182022

Time (h)

Biom

ass (

mM

)






119894= 120593lowast





0 50 100 15015

2

25

3

35

4

Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)

Glu

cose

(mM

)

0

05

1

15

2

25

3

35

Glu

tam

ine (

mM

)

01

02

03

04

05

06

Asp

arag

ine (

mM

)

0

005

01

015

02

025

03

Asp

arta

te (m

M)

0 50 100 150Time (h)










50 100 150

05

1

15

2

25

Time (h)

Lact

ate (

mM

)

04

06

08

1

12

14

16

18

2

Ala

nine

(mM

)

03

04

05

06

07

Prol

ine (

mM

)

0

1

2

3

4

0 50 100 150Time (h)

0

50 100 150Time (h)

050 100 150Time (h)

0

times10minus4





Ant

ibod

y (10eminus4

mM

)








0

05

1

15

2

25

Am

mon

ia (m

M)

0

005

01

015

02

025G

luta

mat

e (m

M)

0

02

04

06

08

1

0

02

04

06

08

1

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)





Viab

le C

ells

Dea

d C

ells (1

0e6

cells

mL)

(10e6

cells

mL)











7 11987011987826 11987011987828 11987011987856

)





120593lowast

2[pmol(cell h)] 00191 00058 112 times 106

120593lowast


120593lowast


120593lowast


120593lowast


120593lowast

7[pmol(cell h)] 06429 00573 136 times 103

120593lowast


120593lowast

9[pmol(cell h)] 00731 00113 183 times 104

11987011987811

[mM] 001 001 163 times 105

11987011987812

[mM] 001 001 108 times 104

11987011987813

[mM] 001 001 144 times 104

11987011987832

[mM] 0001 0001 964 times 102

11987011987833

[mM] 0001 0001 104 times 103

11987011987834

[mM] 0001 0001 542 times 10minus2

11987011987826

[mM] 001 001 303 times 103

11987011987827

[mM] 001 001 639 times 103

11987011987828

[mM] 001 001 373 times 10minus1

11987011987829

[mM] 001 001 323 times 105

11987011987845

[mM] 0001 0001 742 times 103

11987011987856

[mM] 0001 0001 445 times 103

120583 00399 00399 322 times 10minus2

119896119889

006 006 499 times 10minus2


4 Conclusions










120593lowast


120593lowast


120593lowast


120593lowast

5[pmol(cell h)] 7873 23

120593lowast


120593lowast


120593lowast


120593lowast


11987011987811

[mM] 8936 times 105 1711987011987812

[mM] 7030 times 104 2911987011987813

[mM] 3684 times 104 511987011987832

[mM] 731 times 102 2911987011987833

[mM] 2565 times 103 1411987011987834

[mM] 0026 1311987011987826

[mM] 2295 times 104 8711987011987827

[mM] 9273 times 103 5111987011987828

[mM] 0649 11611987011987829

[mM] 6224 times 105 3911987011987845

[mM] 1814 times 103 4411987011987856

[mM] 4657 times 104 75120583 0048 2119896119889

0070 073

Acknowledgments


References




































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology





120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


120593lowast


11987011987811

[mM] 8989 times 105

11987011987812

[mM] 6495 times 104

11987011987813

[mM] 3723 times 104

11987011987832

[mM] 7076 times 102

11987011987833

[mM] 2782 times 103

11987011987834

[mM] 001911987011987826

[mM] 2719 times 104

11987011987827

[mM] 9324 times 103

11987011987828

[mM] 053711987011987829

[mM] 6683 times 105

11987011987845

[mM] 1920 times 103

11987011987856

[mM] 4488 times 104

120583 0043119896119889

0067

st 12058510((119896 + 1) sdot 119879

119904)

=12058510(119896 sdot 119879119904) + 119879119904sdot

120593lowast


11987011987827

+ 1205852(119896 sdot 119879119904)

(17)



7and119870





0 50 100 15002468

10121416182022

Time (h)

Biom

ass (

mM

)






119894= 120593lowast





0 50 100 15015

2

25

3

35

4

Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)

Glu

cose

(mM

)

0

05

1

15

2

25

3

35

Glu

tam

ine (

mM

)

01

02

03

04

05

06

Asp

arag

ine (

mM

)

0

005

01

015

02

025

03

Asp

arta

te (m

M)

0 50 100 150Time (h)










50 100 150

05

1

15

2

25

Time (h)

Lact

ate (

mM

)

04

06

08

1

12

14

16

18

2

Ala

nine

(mM

)

03

04

05

06

07

Prol

ine (

mM

)

0

1

2

3

4

0 50 100 150Time (h)

0

50 100 150Time (h)

050 100 150Time (h)

0

times10minus4





Ant

ibod

y (10eminus4

mM

)








0

05

1

15

2

25

Am

mon

ia (m

M)

0

005

01

015

02

025G

luta

mat

e (m

M)

0

02

04

06

08

1

0

02

04

06

08

1

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)





Viab

le C

ells

Dea

d C

ells (1

0e6

cells

mL)

(10e6

cells

mL)











7 11987011987826 11987011987828 11987011987856

)





120593lowast

2[pmol(cell h)] 00191 00058 112 times 106

120593lowast


120593lowast


120593lowast


120593lowast


120593lowast

7[pmol(cell h)] 06429 00573 136 times 103

120593lowast


120593lowast

9[pmol(cell h)] 00731 00113 183 times 104

11987011987811

[mM] 001 001 163 times 105

11987011987812

[mM] 001 001 108 times 104

11987011987813

[mM] 001 001 144 times 104

11987011987832

[mM] 0001 0001 964 times 102

11987011987833

[mM] 0001 0001 104 times 103

11987011987834

[mM] 0001 0001 542 times 10minus2

11987011987826

[mM] 001 001 303 times 103

11987011987827

[mM] 001 001 639 times 103

11987011987828

[mM] 001 001 373 times 10minus1

11987011987829

[mM] 001 001 323 times 105

11987011987845

[mM] 0001 0001 742 times 103

11987011987856

[mM] 0001 0001 445 times 103

120583 00399 00399 322 times 10minus2

119896119889

006 006 499 times 10minus2


4 Conclusions










120593lowast


120593lowast


120593lowast


120593lowast

5[pmol(cell h)] 7873 23

120593lowast


120593lowast


120593lowast


120593lowast


11987011987811

[mM] 8936 times 105 1711987011987812

[mM] 7030 times 104 2911987011987813

[mM] 3684 times 104 511987011987832

[mM] 731 times 102 2911987011987833

[mM] 2565 times 103 1411987011987834

[mM] 0026 1311987011987826

[mM] 2295 times 104 8711987011987827

[mM] 9273 times 103 5111987011987828

[mM] 0649 11611987011987829

[mM] 6224 times 105 3911987011987845

[mM] 1814 times 103 4411987011987856

[mM] 4657 times 104 75120583 0048 2119896119889

0070 073

Acknowledgments


References




































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


0 50 100 15015

2

25

3

35

4

Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)

Glu

cose

(mM

)

0

05

1

15

2

25

3

35

Glu

tam

ine (

mM

)

01

02

03

04

05

06

Asp

arag

ine (

mM

)

0

005

01

015

02

025

03

Asp

arta

te (m

M)

0 50 100 150Time (h)










50 100 150

05

1

15

2

25

Time (h)

Lact

ate (

mM

)

04

06

08

1

12

14

16

18

2

Ala

nine

(mM

)

03

04

05

06

07

Prol

ine (

mM

)

0

1

2

3

4

0 50 100 150Time (h)

0

50 100 150Time (h)

050 100 150Time (h)

0

times10minus4





Ant

ibod

y (10eminus4

mM

)








0

05

1

15

2

25

Am

mon

ia (m

M)

0

005

01

015

02

025G

luta

mat

e (m

M)

0

02

04

06

08

1

0

02

04

06

08

1

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)





Viab

le C

ells

Dea

d C

ells (1

0e6

cells

mL)

(10e6

cells

mL)











7 11987011987826 11987011987828 11987011987856

)





120593lowast

2[pmol(cell h)] 00191 00058 112 times 106

120593lowast


120593lowast


120593lowast


120593lowast


120593lowast

7[pmol(cell h)] 06429 00573 136 times 103

120593lowast


120593lowast

9[pmol(cell h)] 00731 00113 183 times 104

11987011987811

[mM] 001 001 163 times 105

11987011987812

[mM] 001 001 108 times 104

11987011987813

[mM] 001 001 144 times 104

11987011987832

[mM] 0001 0001 964 times 102

11987011987833

[mM] 0001 0001 104 times 103

11987011987834

[mM] 0001 0001 542 times 10minus2

11987011987826

[mM] 001 001 303 times 103

11987011987827

[mM] 001 001 639 times 103

11987011987828

[mM] 001 001 373 times 10minus1

11987011987829

[mM] 001 001 323 times 105

11987011987845

[mM] 0001 0001 742 times 103

11987011987856

[mM] 0001 0001 445 times 103

120583 00399 00399 322 times 10minus2

119896119889

006 006 499 times 10minus2


4 Conclusions










120593lowast


120593lowast


120593lowast


120593lowast

5[pmol(cell h)] 7873 23

120593lowast


120593lowast


120593lowast


120593lowast


11987011987811

[mM] 8936 times 105 1711987011987812

[mM] 7030 times 104 2911987011987813

[mM] 3684 times 104 511987011987832

[mM] 731 times 102 2911987011987833

[mM] 2565 times 103 1411987011987834

[mM] 0026 1311987011987826

[mM] 2295 times 104 8711987011987827

[mM] 9273 times 103 5111987011987828

[mM] 0649 11611987011987829

[mM] 6224 times 105 3911987011987845

[mM] 1814 times 103 4411987011987856

[mM] 4657 times 104 75120583 0048 2119896119889

0070 073

Acknowledgments


References




































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


50 100 150

05

1

15

2

25

Time (h)

Lact

ate (

mM

)

04

06

08

1

12

14

16

18

2

Ala

nine

(mM

)

03

04

05

06

07

Prol

ine (

mM

)

0

1

2

3

4

0 50 100 150Time (h)

0

50 100 150Time (h)

050 100 150Time (h)

0

times10minus4





Ant

ibod

y (10eminus4

mM

)








0

05

1

15

2

25

Am

mon

ia (m

M)

0

005

01

015

02

025G

luta

mat

e (m

M)

0

02

04

06

08

1

0

02

04

06

08

1

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)





Viab

le C

ells

Dea

d C

ells (1

0e6

cells

mL)

(10e6

cells

mL)











7 11987011987826 11987011987828 11987011987856

)





120593lowast

2[pmol(cell h)] 00191 00058 112 times 106

120593lowast


120593lowast


120593lowast


120593lowast


120593lowast

7[pmol(cell h)] 06429 00573 136 times 103

120593lowast


120593lowast

9[pmol(cell h)] 00731 00113 183 times 104

11987011987811

[mM] 001 001 163 times 105

11987011987812

[mM] 001 001 108 times 104

11987011987813

[mM] 001 001 144 times 104

11987011987832

[mM] 0001 0001 964 times 102

11987011987833

[mM] 0001 0001 104 times 103

11987011987834

[mM] 0001 0001 542 times 10minus2

11987011987826

[mM] 001 001 303 times 103

11987011987827

[mM] 001 001 639 times 103

11987011987828

[mM] 001 001 373 times 10minus1

11987011987829

[mM] 001 001 323 times 105

11987011987845

[mM] 0001 0001 742 times 103

11987011987856

[mM] 0001 0001 445 times 103

120583 00399 00399 322 times 10minus2

119896119889

006 006 499 times 10minus2


4 Conclusions










120593lowast


120593lowast


120593lowast


120593lowast

5[pmol(cell h)] 7873 23

120593lowast


120593lowast


120593lowast


120593lowast


11987011987811

[mM] 8936 times 105 1711987011987812

[mM] 7030 times 104 2911987011987813

[mM] 3684 times 104 511987011987832

[mM] 731 times 102 2911987011987833

[mM] 2565 times 103 1411987011987834

[mM] 0026 1311987011987826

[mM] 2295 times 104 8711987011987827

[mM] 9273 times 103 5111987011987828

[mM] 0649 11611987011987829

[mM] 6224 times 105 3911987011987845

[mM] 1814 times 103 4411987011987856

[mM] 4657 times 104 75120583 0048 2119896119889

0070 073

Acknowledgments


References




































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


0

05

1

15

2

25

Am

mon

ia (m

M)

0

005

01

015

02

025G

luta

mat

e (m

M)

0

02

04

06

08

1

0

02

04

06

08

1

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)

0 50 100 150Time (h)





Viab

le C

ells

Dea

d C

ells (1

0e6

cells

mL)

(10e6

cells

mL)











7 11987011987826 11987011987828 11987011987856

)





120593lowast

2[pmol(cell h)] 00191 00058 112 times 106

120593lowast


120593lowast


120593lowast


120593lowast


120593lowast

7[pmol(cell h)] 06429 00573 136 times 103

120593lowast


120593lowast

9[pmol(cell h)] 00731 00113 183 times 104

11987011987811

[mM] 001 001 163 times 105

11987011987812

[mM] 001 001 108 times 104

11987011987813

[mM] 001 001 144 times 104

11987011987832

[mM] 0001 0001 964 times 102

11987011987833

[mM] 0001 0001 104 times 103

11987011987834

[mM] 0001 0001 542 times 10minus2

11987011987826

[mM] 001 001 303 times 103

11987011987827

[mM] 001 001 639 times 103

11987011987828

[mM] 001 001 373 times 10minus1

11987011987829

[mM] 001 001 323 times 105

11987011987845

[mM] 0001 0001 742 times 103

11987011987856

[mM] 0001 0001 445 times 103

120583 00399 00399 322 times 10minus2

119896119889

006 006 499 times 10minus2


4 Conclusions










120593lowast


120593lowast


120593lowast


120593lowast

5[pmol(cell h)] 7873 23

120593lowast


120593lowast


120593lowast


120593lowast


11987011987811

[mM] 8936 times 105 1711987011987812

[mM] 7030 times 104 2911987011987813

[mM] 3684 times 104 511987011987832

[mM] 731 times 102 2911987011987833

[mM] 2565 times 103 1411987011987834

[mM] 0026 1311987011987826

[mM] 2295 times 104 8711987011987827

[mM] 9273 times 103 5111987011987828

[mM] 0649 11611987011987829

[mM] 6224 times 105 3911987011987845

[mM] 1814 times 103 4411987011987856

[mM] 4657 times 104 75120583 0048 2119896119889

0070 073

Acknowledgments


References




































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology





120593lowast

2[pmol(cell h)] 00191 00058 112 times 106

120593lowast


120593lowast


120593lowast


120593lowast


120593lowast

7[pmol(cell h)] 06429 00573 136 times 103

120593lowast


120593lowast

9[pmol(cell h)] 00731 00113 183 times 104

11987011987811

[mM] 001 001 163 times 105

11987011987812

[mM] 001 001 108 times 104

11987011987813

[mM] 001 001 144 times 104

11987011987832

[mM] 0001 0001 964 times 102

11987011987833

[mM] 0001 0001 104 times 103

11987011987834

[mM] 0001 0001 542 times 10minus2

11987011987826

[mM] 001 001 303 times 103

11987011987827

[mM] 001 001 639 times 103

11987011987828

[mM] 001 001 373 times 10minus1

11987011987829

[mM] 001 001 323 times 105

11987011987845

[mM] 0001 0001 742 times 103

11987011987856

[mM] 0001 0001 445 times 103

120583 00399 00399 322 times 10minus2

119896119889

006 006 499 times 10minus2


4 Conclusions










120593lowast


120593lowast


120593lowast


120593lowast

5[pmol(cell h)] 7873 23

120593lowast


120593lowast


120593lowast


120593lowast


11987011987811

[mM] 8936 times 105 1711987011987812

[mM] 7030 times 104 2911987011987813

[mM] 3684 times 104 511987011987832

[mM] 731 times 102 2911987011987833

[mM] 2565 times 103 1411987011987834

[mM] 0026 1311987011987826

[mM] 2295 times 104 8711987011987827

[mM] 9273 times 103 5111987011987828

[mM] 0649 11611987011987829

[mM] 6224 times 105 3911987011987845

[mM] 1814 times 103 4411987011987856

[mM] 4657 times 104 75120583 0048 2119896119889

0070 073

Acknowledgments


References




































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology





120593lowast


120593lowast


120593lowast


120593lowast

5[pmol(cell h)] 7873 23

120593lowast


120593lowast


120593lowast


120593lowast


11987011987811

[mM] 8936 times 105 1711987011987812

[mM] 7030 times 104 2911987011987813

[mM] 3684 times 104 511987011987832

[mM] 731 times 102 2911987011987833

[mM] 2565 times 103 1411987011987834

[mM] 0026 1311987011987826

[mM] 2295 times 104 8711987011987827

[mM] 9273 times 103 5111987011987828

[mM] 0649 11611987011987829

[mM] 6224 times 105 3911987011987845

[mM] 1814 times 103 4411987011987856

[mM] 4657 times 104 75120583 0048 2119896119889

0070 073

Acknowledgments


References




































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology












































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology

mammalian cell culture process for monoclonal …...mammalian cell culture process for monoclonal...

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