64

UDCUDCUDCUDC 577.31577.31577.31577.31 :::: 662.754662.754662.754662.754 =111=111=111=111 ScientificScientificScientificScientific paperpaperpaperpaper

THETHETHETHEKINETICKINETICKINETICKINETICMODELSMODELSMODELSMODELS OFOFOFOF THETHETHETHE BIOPROCESSBIOPROCESSBIOPROCESSBIOPROCESSWITHWITHWITHWITH FREEFREEFREEFREE ANDANDANDANDIMMOBILIZEDIMMOBILIZEDIMMOBILIZEDIMMOBILIZED CELLSCELLSCELLSCELLS

StankoStankoStankoStanko ŽŽŽŽerajierajierajierajićććć1111,,,, JelenkaJelenkaJelenkaJelenka SavkoviSavkoviSavkoviSavkovićććć-Stevanovi-Stevanovi-Stevanovi-Stevanovićććć22221Faculty of Technology, Leskovac, Serbia

2Faculty of Technology and Metallurgy, Belgrade, Serbia

The kinetic models are fundamental knowledge about biochemical and microbialprocesses. The object of the work is the kinetic model researching based on thebiochemical and microbial mechanisms of cell growth, substrate consumption andproduct formation. The ethanol biosynthesis process as fuel from renewable resourcesis the engineering goal. The engineering goal requires the inclusion of the inhibitoryeffect of high substrate and ethanol concentration in the kinetic models. Theinteractions of the ethanol synthesis with growth biomass mechanism were researched.The models of the ethanol forming mechanisms were incorporated in the complexstructure of the kinetic models. The researches of the kinetic models were realized bycomputer simulation and experimental verification at the ethanol biosynthesis processfrom glucose with free and immobilized cells S. cerevisiae. The model parameters andlimits were estimated by experimental validation. The base of the verified deterministicand semi-empiric kinetic models was created. The model base may be onward used forthe process simulation models generating.KeyKeyKeyKey words:words:words:words: Kinetic model, ethanol biosynthesis, free and immobilized cells, modelstructured and validation.

INTRODUCTIONINTRODUCTIONINTRODUCTIONINTRODUCTION

Mathematical models based on physical and chemical laws (e.g., mass and energybalances, thermodynamics, chemical reaction kinetics) are frequently employed inprocess description. These models are conceptually attractive because a general modelfor any system size can be developed, even before the system is constructed. On theother hand, an empirical model can be devised which simply correlates input-outputdata without any physicochemical analysis of the process [1].

A very important kinetic equation according to Monod gives an expression for thegrowth rate of an organism when growth is limited by one substrate. The Monodequation is analogous to the Briggs-Haldane solution of the Michaelis-Menten modelfor the kinetics of a single enzyme. If growth is considered to be the result of asequence of enzymatic reactions in which one reaction is much slower than all others,the Michaelis-Menten equation can be good fit, which the Monod equation often gives

65

in modeling growth. This reasoning is, however, by no means unique. An equation ofthe form of basic Monod equation is obtained starting from various other modelingassumptions [2].

Although the growth of microorganism is an unusually comlex phenomenon, it is oftenpossible to represent this growth by relatively simple laws [3]. J. Von Liebig’s (1840)pioneering studies are in the base of the idea of the substrate-limiting cell growth. Heformulated the Law of the Minimum, which described the effect of essential nutrientson cell growth. It states that growth is controlled not by the total of resources available,but by the scarcest resource. This concept was originally applied to plants. The growthof an organism may be dependent on a number of different factors. The availability ofthese may vary, such that at any given time one is more limiting than the others.Liebig's Law states that growth only occurs at the rate permitted by the most limiting.Liebig's Law has been extended to description the rates of enzimatic reactions(Michaelis-Menten [4]) and growth biological populations (Monod [5,6,7,8]).

Some authors [9] discarded the conclusion that Monod proposed the equation ontheoretical base, indicating the major difference between the Monod

equation for microbial growth and the Michaelis-Menten equation

for enzymatic reaction lies on the process state, i.e. equilibrium for

enzymatic reaction and non-equilibrium for microbial growth process. Anumber of author has pointed out that micro-organisms, from a view-point ofthermodynamics, are to be considered as open systems, which is not in a state ofthermodynamic equilibrium [10], but in the dynamic equilibrium. The dynamicequilibrium is steady state in which there are continuous reactant input andtransformations of reaction product.The production of fuel ethanol from renawable resources has initiated the research ofthe kinetic model based on the biochemical and microbial mechanisms of cell growth,substrate consumption and product formation. In this paper, the live growing cells wereopen system, which is characterized by dynamic equilibrium. The steady stateconcentrations in the dynamic equilibrium are different than the thermodynamic steadystate concentrations. Therefore the reaction progress is permanent according toequilibrium. A microorganism for maintenance and growth consumes the energy,which is generated thereat.

FreeFreeFreeFree suspendedsuspendedsuspendedsuspended andandandand immobilizedimmobilizedimmobilizedimmobilized cellscellscellscells

Microbiological transformation processes are the natural state of live and growing free-suspended cells. At batch bioprocesses, they are limited by low cell concentration andshort life half time. Multiple use required additional separation subsystem. Continualsystems are limited by cell washing from reactors at rate higher than specific growthrate. A fed batch technique partionally solves the problem of low cell concentration.

66

The processed with recirculation do that as well. In all bioprocesses with free cells,volumetric productivity is relatively small because of inhibitory effect of highconcentration of substrate and product to the cell growth.Immobilized systems provide high cell concentration in the process, high productivity,better control and incomparably higher operational stability. There are two approachesin the development of immobilized cells.

The first one is the immobilized biocatalyst design without or with minimum cellwashing. The goal is the high starting cell concentration which is maintained constantall through the process by effect of matrix, two-layer and lowering nutrientconcentration which is limiting for the growth, and is not a substrate e.g. nitrogen. Avery high operational stability is achieved that way. The technique of immobilizationin two-layer particles additionally decreases cell washing and increases operationalbiosystem stability [11].

Immobilized biocatalyst designed with cell washing, allows limited

growth of cells that are washed from the matrix. This technique is

used at production of starter cultures. Thereat, immobilized and

washed cells, which function as free, increase the total effect on

productivity. However, washed cells and non-controlled growth in the

matrix lead to matrix destruction with exploitation time, thus

decreasing operational biosystem stability.

MATERIALSMATERIALSMATERIALSMATERIALS ANDANDANDANDMETHODSMETHODSMETHODSMETHODS

As a case study, the anaerobic ethanol fermentation process with free suspended andimmobilized S. cerevisiae cells was used [12]. The nutritive medium content, (g/lit):extract yeast 10.0, NaCl 1.0, CaCl22H2O 0.2, KH2PO4 2.0, FeCl36H2O 0.01,MgSO47H2O 1.7, NH4Cl 2.0, C6H12O6 20.0. The nutrient medium was prepared andsterilized in citrate puffer in pH 5. The nutritive medium content was optimized inobject function of maximum rate of cells growth. The kinetic measurements wererealized in temperature 320C.For free suspended cells, the kinetic experimental data are gained on the batchbioprocess with initial substrate concentration cS0=100-250 g/dm3 and inoculumscX0=0,75 g/dm3, in well-mixed process vessel by method of initial rate.For immobilized cells in Ca-alginate gel bead, the experimental data are gained on thebatch process with well mixing at initial substrate concentration cS0=100-250 g/dm3and very low immobilized cells concentration in the free volume of the process vessel

67

cX0=3,50 g/dm3. In the goal of elimination the restricting effect on the bioprocesskinetics, the phenomena of convection, inner and interphase transfer of the glucosesubstrate mass and ethanol production are researched experimentally, by increasing therecirculation-mixing rate. The bioprocess effectiveness was determined by theindependent kinetic and diffusion experiment. The effectiveness of immobilizedbiocatalyst is determined by the effective diffusion coefficient, which is modeled andconfirmed on the experimental data [13].The process model is verified on the experimental data obtained on the flow hemostatand column with bed immobilized biocatalyst. The column parameters are porosityε =0.51, immobilized cells concentration in void volume of the column cX=163.33g/dm3 and the recirculation flow that enables well-mixed system.

MODELSMODELSMODELSMODELS DEVELOPEMENTDEVELOPEMENTDEVELOPEMENTDEVELOPEMENT

The mathematical expressions describing the rates of biomass and product formationand the rates of substrate utilization are based on the Monod’s model and its derivatedforms. It is seeking to describe both free and immobilized cells ethanol fermentationwith simple model. The Monod’s model has been modified to include termsrepresenting maintenance coefficient as well as the effects of product and substrateinhibition.The maintenance processes, which have to do with the energy which has to be suppliedto maintain the system, even if growth is absent. The maintenance requirement isassumed to be a constant independent of the growth rate of the cells.The kinetics of the microbial populations can be configured so as to include inhibitionphenomena, growth associated and non-growth associated patterns of productformation. The system can be structured to contain either one or two differentmicrobial populations or the diauxic utilization of multiple substrates as differentcarbon-energy sources. The two populations can interact through competition,commensalisms, cross-inhibition or predation or through a combination of theserelationships.

TheTheTheThe structurestructurestructurestructure ofofofof kinetickinetickinetickinetic modelmodelmodelmodel forforforfor bioethanolbioethanolbioethanolbioethanol synthesissynthesissynthesissynthesis

The structure of kinetic model of isotherm anaerobic bioprocess consists of the modelof biomass growth kinetics and of the models of substrate consumption and productforming kinetics. The models of biomass growth generated on the basic of themechanism and stoichiometrics are analyzed. At the same time, the models on the basisof substrate consumption and product forming pattern are analyzed. For each generatedmodel of the biomass growth kinetics, the determined number of structures ofbioprocess kinetic models is formed including models of different mechanisms of thepatterns of substrate to product conversion.

TheTheTheThe modelsmodelsmodelsmodels ofofofof biomassbiomassbiomassbiomass growthgrowthgrowthgrowth kineticskineticskineticskinetics

68

In this paper the eight kinetic models of biomass growth were generated. The fourmodels represent deterministic models, generated from mechanisms and stoichiometric.These models include different inhibitory effects of high substrate and productconcentration and effects of maintenance to the cell growth (MODEL 1-4) [13].Mixed models (MODEL 5-8) represent semi-empirical models. A part of these modelsis deterministic, multiplied by empirical members for growth inhibition by highsubstrate and product concentration. This approach allows mixed semi-empiricalmodels generation, and they include boundary values of substrate, product ormetabolite concentration that stop cell growth or biosynthesis. It is also possible todescribe biological switching, like pattern changing at second product forming, diauxicprocess or processes with two different microbial population.

MODEL 1. Monod’s growth biomass model for single-substrate limitation,

SM

SmX

XX cK

ccdt

dcr+⋅

⋅==µ

(1)

MODEL 2. Monod’s biomass growth model, which includes maintenance process,

XSM

SmX

XX cmcK

ccdt

dcr ⋅−+⋅

⋅==µ

(2)

MODEL 3. Biomass growth model, which includes non-competitive substrate andproduct inhibition,

X

IP

PS

IS

SSM

SmX

XX cm

Kcc

KccK

ccdt

dcr ⋅−⋅

+++

⋅⋅== 2

µ(3)

MODEL 4. Biomass growth model, which includes non-competitive substrate andcompetitive product inhibition,

X

IS

S

IP

PMSM

SmX

XX cm

Kc

KcKcK

ccdt

dcr ⋅−+++

⋅⋅== 2

µ(4)

MODEL 5. Mixed semi-empirical biomass growth model, which includes modifiednon-competitive substrate inhibition and linear product inhibition,

Xm

P

IS

SSM

SmX

XX cmP

c

KccK

ccdt

dcr ⋅−⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅

++

⋅⋅== 12

µ(5)

69

MODEL 6. Mixed semi-empirical biomass growth model, which

includes modified non-competitive substrate and product inhibition

and linear product inhibition,

Xm

P

IP

PSP

P

IS

SSM

SmX

XX cmP

c

KccK

c

KccK

ccdt

dcr ⋅−⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅

++⋅

++

⋅⋅== 122

µ(6)

MODEL 7. Mixed semi-empirical biomass growth model, which

includes multiplicated of modified non-competitive substrate and

product inhibition,

X

IP

PSP

P

IS

SSM

SmX

XX cm

KccK

c

KccK

ccdt

dcr ⋅−++

⋅

++

⋅⋅== 22

µ(7)

MODEL 8. Modification of the Levenspiel’s semi-empirical model

which includes basic Monod’s model extended by substrate and

product inhibition,

X

n

m

Sn

m

P

SM

SmX

XX c m - S

c - 1Pc - 1

c + Kc c =

dtdcr

SP

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅

⋅⋅=µ (8)

At bioprocess with alive and growing cells the molecular mass of cell may not exactlybe defined and determined. The partial knowledge of biomass average compositionmay not link with effective molar concentration. Also, do to cells growth and deathwith time, the biomass concentration changes.

By using analogy with enzymatic catalysis theory, which is based in dynamic balance,the microbial process with live and growing cell may be described by modifiedMonod’s growth model [8,9], by transforming Eq. (1),

21 m SX X

S S PX XM S

IS IP

cr dc mc c cc c dt K cK K

µµ

⋅= = = −

⋅+ + +

(9)

If molar concentration Sc and Pc , Michaelis’ constant MK and inhibitory constants

ISK and IPK are, multiplied with adequate molar mass of substrate SM , product PM ,

70

and average molar mass of cells XM , the dimensional correct Monod’s equation withspecific growth rate µ in units of specific mass rate ( X Xg g h ) is obtained [9].

Regardless of the complexity of biological mechanism in the growing cell, in the finalinstance, the transformation processes of the substrate into the product are taking placeon the enzyme. If the constant part taking of the enzyme in growing cells wasapproximately supposed, then the concentration of cell biomass in the bioprocesswould follow the appropriate change of enzyme concentration in the mol and in themass units. This approach is necessary considering the fact that the experimental dataare gained for cell biomass in the units of mass concentration during the microbeprocess model validation.The non-structured model, that does not take the inner microorganism structure, is usedat bioprocess system modeling. The partial structuring is related to the enzyme in thecontent of cell biomass. The model is also distributed because the condition of theculture is characterized by the total biomass concentration Xc (gSM/dm3). If thestructure of cell biomass is supposed to be averagely constant and independent fromthe cell age, in that case Monod’s non-structured and distributed model may beconsidered theoretical.

KineticKineticKineticKinetic modelsmodelsmodelsmodels ofofofof thethethethe substratesubstratesubstratesubstrate consumptionconsumptionconsumptionconsumption

The substrate requirements for the increase in biomass can be considered to becomposed of two contributions: the substrate requirements for the synthesis of theprecursors for biomass, which can be calculated from the atomic conservation principle,and the substrate requirements for biomass synthesis from precursors.The organisms take up substrates, such as a carbohydrate, oxygen, a nitrogen sourceetc., and convert them to cellular mass, products (for example ethanol) and metabolites(water, carbon dioxide etc.). From the principle of conservation of atomic species ageneralized stoichiometric equation can be derived, which could take the form:

l m n j j a b cj

CH O N A CH O Nα β+ ⎯⎯→∑ (10)in which

l m nCH O N is the composition formula of the main energy supplying substrate,

jA are other substrate metabolite and products (O2, CO2, nitrogen source),

j ,α β are stoichiometric coefficients, and

a b cCH O N is the composite formula of the micro-organisms.

On average 8% of the biomass is composed of elements other than C, H, O and N, sofor calculation of actual requirements 8% has to be deducted from the values calculatedhere [2]. In anaerobic condition the oxygen source is glucose substrate. This decreasesbiomass yield and increases ethanol yield. If the substrate is glucose, which is nitrogen

71

non-content, then it is necessary to add nitrogen source as another substrate (term Aj inEq.10).Many authors researched initial nitrogen concentration effect on

maximal specific growth rate and maximal production rate of

ethanol [14,15]. This researches confirmed basic Monod’s model of the

substrate-nitrate limiting cell growth. X Xr cµ= , (11)

where X N X Nr r Y= ⋅ and N cm

N NK cµ

µ =+

(12)

and parameters values: 0 294m .µ = , NK =0.1 and YX/N=6.84.Initial nitrogen concentration influence was investigated. Model validity is confirmedand maximal specific rate was gained at 0.2 gN/dm3.Their conclusion is in good agreement with that of Saita and Slaughter [16], whofound that nitrogen has a stimulating effect on the rate of fermentation trough itsfunction as a substrate for protein synthesis, and not by an activation of glycolyticenzymes by ammonium ions.The nitrogen influences itself in the cell growth, especially for the synthesis of theprecursor for biomass, and it is may be used for control of the undesirable biomassgrowth at immobilized cells.If the substrate is simultaneous carbon source for cells growth, reactant whenconverted to the ethanol, energy source for biotransformation and source formaintenance energy, then a balance of substrate consumption is as following,

S S S S S

biomass product growth energy maintenance energy

dc dc dc dc dcdt dt dt dt dt

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(13)

At bioprocess synthesis with immobilized cells, the balance (Eq. 13)

was reduced,

S S S

product maintenance energy

dc dc dcdt dt dt

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(14)

In restrictive condition, e.g. drastically growth limitation with glucose or nitrogen,minimum produced energy is consumed primarily as maintenance energy andsecondary as growth cells energy.

KineticKineticKineticKinetic modelsmodelsmodelsmodels ofofofof thethethethe productproductproductproduct synthesissynthesissynthesissynthesis

Bioprocesses are, in general, integrated biochemical, physiological andphysicochemical component. The simultaneous solving of biomass growth model, with

72

the models of substrate consumption and product forming, enables valid simulation ofreal bioprocess dynamics.The additional analyses are directed to the interaction of biochemical component with abiological one. The enzymatic bioprocesses are different than the bioprocesses ingrowing cells in the dynamics of the enzymatic complex quantity increase at growingcells [6]. Therefore comes the idea that the additional phenomenon, which should beanalyzed and modeled, is the phenomenon of the biotransformation interaction with thecell growth. The tree basic mechanisms of product forming are chosen for the analysisand modeling i.e. growth-associated, non-growth-associated or metabolically non-coupling product forming and non-growth associated forming of the secondarymetabolite [7].The modeling of these mechanisms leads to the transformation in the kinetic models ofproduct forming and substrate consumption. For all that, the kinetic model of biomassgrowth stays unchanged.Biochemical kinetic models are developed and they include modified Monod’sbiomass growth model for the processes inhibited by the high substrate and productconcentration, by mechanistic and empiric terms. The product forming and substrateconsumption models are generated by the three elected and supposed product-formingmechanisms.For growth associated product formation (Eqs. 15-17), the rates of product formationand the rate of substrate utilization are specified as stoichiometrically proportional to(ie. associated with) the biomass growth rate.Non- growth associated product formation as a result of metabolic uncoupling, issimulated by specifying that only the biomass growth rate is subject to substrate andproduct inhibition (Eq. 22). Product formation is linked to substrate utilization andceases when the substrate runs out. Product formation is thus not directly associatedwith biomass growth and substrate utilization is not coupled with growth (Eqs. 23-24).In the absence of inhibition, the product formation pattern is growth associated. Themodel is thus consistent with experimental observations of metabolic uncoupling.When non-growth associated product formation is modelled as a phenomenonassociated with secondary metabolite formation, the rate of product formation is linkedto the endogenous rate of cellular degradation (ie. endogenous metabolism). Productformation is thus a process that is secondary to biomass growth. In addition, productformation but not growth is inhibited by the concentration of the substrate. Thisinhibition pattern simulates that observed with the production of a wide variety ofsecondary metabolites.This approach is in many ways superior to the mathematical model, which has mostoften been used to describe growth and non-growth associated product formation bythe “mixed-growth-associated” model of Leudeking and Piret [17].Unlike Leudeking and Piret's model, the models generated by this approach recognizethe fact that there are many reasons behind non-growth associated patterns of productformation; e.g. metabolic uncoupling and secondary metabolite formation. In contrast,Leudeking and Piret's model is a curve fitting relationship, which does not differentiatebetween the different mechanisms behind non-growth associated product formation. In

73

addition, in this models, product and biomass formation are stoichiometricallybalanced with substrate utilization. This is not the case with Leudeking and Piret'smodel, which is unable to predict a cessation of product formation.The modeling methodology of microbial kinetics is represented in detail on the modelsstructured according to the biomass growth model with the non-competitive substrateand product inhibition (MODEL 3), coupled with the models of substrate consumptionand models of growth and non-growth associated product formation.

GrowthGrowthGrowthGrowth associatedassociatedassociatedassociated modelmodelmodelmodel productproductproductproduct formationformationformationformation

In the growth-associated model product forming (MODEL 3.1), which includes growthbiomass model with non-competitive substrate and product inhibition Eqs. (15-17), thetechnique formed of the kinetic model structure for bioprocess with free cells (MODEL3.1.1) and bioprocess with immobilized cells (MODEL 3.1.2) is presented.

2m SX

X X XS S P

M SIS IP

cdcr c m cc c cdt K cK K

µ ⋅= = ⋅ − ⋅

⋅+ + +

(15)

2S ms S

S XS S P

M SIS IP

dc cr cc c cdt K cK K

ν ⋅− = − = ⋅

⋅+ + +

(16)

2mp SP

P XS S P

M SIS IP

cdcr cc c cdt K cK K

ν ⋅= = ⋅

⋅+ + +

(17)

The bioprocess kinetic model (MODEL 3.1.2) with immobilized cell Eqs. (18-20),which includes growth-associated model product forming, the model Eqs. (15-17) wasaccomplished under presumption that cells concentration is constant,

0==dt

dcr XX (18)

2S ms S

S XS S P

M SIS IP

dc cr cc c cdt K cK K

ν ⋅− = − = ⋅

⋅+ + +

(19)

2mp SP

P XS S P

M SIS IP

cdcr cc c cdt K cK K

ν ⋅= = ⋅

⋅+ + +

(20)

The basic postulate of immobilized system Eq. (21), as bioprocess system in which thespecific biomass growth rate µ is equal to the maintenance coefficient m , followsfrom the condition Eq. (18),

74

2m S

X XS S P

M SIS IP

cc m cc c cK cK K

µ ⋅⋅ = ⋅

⋅+ + +

(21)

The model of a live non-growth cell may approximate this system ( mµ = ). In Eq. (21)maintenance coefficient m represents the energy demands just to maintain the activityin biomass. This condition may be got by control of the nitrogen concentration innutrient medium, following Eq. (12). In other approach the specific growth rate may beexpressed as a function of two limiting nutrient, main substrate glucose and growthlimiting nutrient nitrogen.

Non-GrowthNon-GrowthNon-GrowthNon-Growth associatedassociatedassociatedassociated modelmodelmodelmodel productproductproductproduct formationformationformationformation

The non-growth-associated model product forming (MODEL 3.2), which is metabolicuncoupling and including model biomass growth with non-competitive substrate andproduct inhibition, is generated in the form of Eqs. (22-24) for free suspended cells(MODEL 3.2.1), by the analogy with the model Eqs. (15-17)

2m SX

X X XS S P

M SIS IP

cdcr c m cc c cdt K cK K

µ ⋅= = ⋅ − ⋅

⋅+ + +

(22)

SM

SmsX

SS cK

ccdt

dcr+⋅

⋅=−=−ν

(23)

SM

SmpX

PP cK

cc

dtdcr

+⋅

⋅==ν

(24)

The kinetic model (MODEL 3.2.2) of bioprocess with immobilized cell for non-growth-associated mechanism product forming Eqs. (25-27), from model Eqs. (22-24)and under presumption that cells concentration is constant, was accomplished,

0==dt

dcr XX (25)

SM

SmsX

SS cK

ccdt

dcr+⋅

⋅=−=−ν

(26)

SM

SmpX

PP cK

cc

dtdcr

+⋅

⋅==ν

(27)

ParameterParameterParameterParameter estimationestimationestimationestimation andandandand modelmodelmodelmodel validationvalidationvalidationvalidation

The experimental data from kinetic experiment with free and immobilized cells wereused for the kinetic parameter estimation.

75

The kinetic model of free cells Eqs. (15-17) which includes growth-associated and themodel Eqs. (22-24) with includes non-growth-associated product forming mechanism,are fitted to the experimental data from bioprocess with free suspended cells.The kinetic model of immobilized cells Eqs. (18-20) which includes growth-associatedand the model Eqs. (25-27) which includes non-growth-associated product formingmechanism, are fitted to the experimental data from bioprocess with immobilized cells.For identification on the model parameters, average values of different processvariables (biomass, glucose and ethanol), for three repeated experiments at differenttime points, were taken. The model parameters were calculated based on ordinary leastsquared (LS) method and assisted by a computer program, which minimized thedeviations between the model prediction and the experimental results.

In this paper the two methods were tested. The first method is based on LS method,which minimized the deviation between the differential equation form model andnumerical differentiation of the experimental data. The Levenberg-Marquardt iterativefitting methods, is used at parameter estimation [18,19].In the second method, for the calculation of the model predictions, the system ofdifferential equations was solved using an integration program based on the Runge-Kutta method of fourth order. The optimization program for the direct search of theminimum of the multivariable function was based on the method of Rosenbrock [20].The minimization criteria in parameter optimization computer program was:

2 2

n k

i j ji j

MINSS W= ∆∑∑ (28)where:

- MINSS represent sum of the square of weighed residues- i and j the number of experimental data points and number of variables,

respectively- jW is statistical weight of each variable ( 1W / σ= ), where σ is variance of

repeat experiments- i j∆ is difference between the model and experimental value

The model validity is statistically estimated by Fisher’s test. Fisher’s numerical valueracF is a measure of deviation of the simulated data in the model to the experimental

data from the process. It is numerically determined as relation of the variancenonadequacy of the model to the experimental data and variance of experimental data,

( )

( )

2

2

221

1

i i

i

mod exp

ad icalc

exp mod exp

i

Y Yn mF

(Y ) Y Yn

σσ

−

− −= =−

−

∑

∑(29)

76

where, expY and modY are experimental and simulation data, n and m are number of

experimental data and model parameters, and expY is mean of experimental data.The Fisher’s test was used for comparative analysis of a more valid models andadoption of the adequat. It is determined for the total model totcalcF and for allcomponents XcalcF ,

ScalcF and

PcalcF partially, by using variance additive principle. For

preliminary comparative analysis and preliminary model validity estimation, thecorrelation coefficient was used.

RESULTSRESULTSRESULTSRESULTS ANDANDANDAND DISCUSIONDISCUSIONDISCUSIONDISCUSION

GrowthGrowthGrowthGrowth associatedassociatedassociatedassociated modelmodelmodelmodel productproductproductproduct formationformationformationformation

The Model 3.1 for free suspended and immobilized cells are fitted to the experimentaldata. The parameters are estimates for different initial substrate concentrations. Theresults of estimation for free system is presented in Table 1, and for immobilizedsystem in Table 2.

Table 1. The parameter estimation of free-system kinetic Model 3.1.1, that includesgrowth-associated mechanism of product forming (cS0=50-250 g/dm3, cX0=0.75 g/dm3,m=0.02 )

0Sc mµ msν mpν MK ISK IPK XcalcFS

calcFP

calcFtot

calcF tabF corr

50 0.362 3.220 1.514 2.276 140.0 37.55 0.160 0.042 0.024 0.226 2.484 1.000100 0.266 2.933 1.361 1.551 357.0 33.15 0.167 0.088 0.051 0.306 2.084 0.999150 0.250 2.826 1.241 9.676 606.1 34.87 0.931 0.671 0.315 1.918 1.841 0.997200 0.205 2.348 0.993 50.740 944.0 30.67 1.560 0.124 0.360 2.044 1.592 0.999250 0.280 2.101 0.819 435.40 9977.0 206.10 10.34 4.540 0.761 15.64 1.385 0.999

Table 2. The parameters estimation of immobilized system kinetic Model 3.1.2, thatincludes growth-associated mechanism of product forming (cX0=3.5 g/dm3 )

0Sc msν mpν MK ISK IPK ScalcFP

calcFtot

calcF tabF corr50 2.813 1.265 105.9 44.19 5.43 0.017 0.019 0.036 1.955 1.000

100 4.718 2.255 106.0 54.78 6.26 0.181 0.171 0.353 1.757 0.998150 8.336 3.900 509.3 73.88 11.7 0.123 0.073 0.195 1.624 1.000200 6.525 2.847 1417.0 4453.00 262.50 0.050 0.028 0.078 1.757 1.000250 4.422 1.733 1652.0 12710.00 444.00 0.110 0.203 0.313 1.642 0.999

Non-GrowthNon-GrowthNon-GrowthNon-Growth associatedassociatedassociatedassociated modelmodelmodelmodel productproductproductproduct formationformationformationformation

The Model 3.2 for free suspended and immobilized cells are fitted to the experimentaldata. The parameters are estimated for different initial substrate concentrations. The

77

results of estimation for free system is presented in Table 3, and for immobilizedsystem in Table 4.

Table 3. The parameter estimation of free-system kinetic Model 3.2.1, that includesnon-growth-associated mechanism of product forming (cS0=50-250 g/dm3, cX0=0.75g/dm3, m=0.02)

0Sc mµ msν mpν MK ISK IPK XcalcF ScalcFP

calcFtot

calcF tabF corr

50 0.352 2.170 1.020 1.716 112.6 43.85 0.022 0.008 0.015 0.045 2.484 1.00

0100 0.231 1.799 0.834 5.382 1269.0 110.50 0.114 0.023 0.053 0.190 2.084 0.999150 0.225 1.641 0.721 16.270 420.3 65.58 0.030 0.142 0.257 0.429 1.841 0.998200 0.170 1.010 0.411 40.070 4019.0 37.48 0.570 0.206 0.322 1.097 1.592 0.998250 0.235 2.101 0.819 435.400 10000.0 206.10 0.567 0.349 0.413 1.330 1.385 0.999

Table 4. The parameter estimation of immobilized system kinetic Model 3.2.2, thatincludes non-growth-associated mechanism of product forming (cX0=3.5 g/dm3 )

0Sc msν mpν MK ScalcFP

calcFtot

calcF tabF corr

50 1.620 0.727 81.73 0.002 0.001 0.003 1.955 1.0

00100 2.769 1.297 155.6 0.001 0.001 0.001 1.757 1.000150 4.486 2.093 383.2 0.011 0.002 0.014 1.624 1.000200 6.318 2.754 1384 0.049 0.028 0.077 1.757 1.000250 4.834 1.892 1815 0.110 0.175 0.285 1.642 0.999

ComparativeComparativeComparativeComparative analysisanalysisanalysisanalysis ofofofof thethethethe growthgrowthgrowthgrowth andandandand non-growthnon-growthnon-growthnon-growth associatedassociatedassociatedassociated modelmodelmodelmodel productproductproductproductformationformationformationformation

Comparative analysis of free-suspended cell kinetic models with growth and non-growth associated product forming, show the dominant influence of inhibition by highglucose and ethanol concentration on cell growth. This is specially illustrated on Figure1, comparing the Figures a2 and b2 i.e. a3 and b3. The same conclusion is gained bycomparative analysis of values of partial and total Fischer’s test in the Table 1 forgrowth associated model and Table 3 for non-growth associated model.

The Fischer’s test (Table 2 and 4) and corresponding comparative analysis of theprogress curves for substrate consumption and product forming (Figure 2), show bettercomparison of experimental and simulated data on the MODEL 3.2.2 as well, for non-growth associated product forming with immobilized cells.

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The kinetic models gained from semi-empirical growth model (MODEL 5-8) are alsotested for growth associated and non-growth associated pattern product forming. Thesemodels show good comparison with experimental data.

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Figure 1. Growth cells, substrate consumption and product formation kinetic curves forfree suspended cells, with growth associated (a) and non-growth associated (b) productforming patterns. The experimental data at different initial glucose concentration 50,100, 150, 200 and 250 g/dm3 (x, +, , ◊, ο) and corresponding curves simulation (⎯)in the MODEL 3.1.1 and MODEL 3.2.1, with parameter values from Table 1 andTable 3.

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Figure 2. Substrate consumption (c1,d1) and product formation (c2,d2) kinetic curvesfor immobilized cells, with growth associated (c) and non-growth associated (d)product forming patterns. The experimental data at different initial glucoseconcentration 50, 100, 150, 200 and 250 g/dm3 (x, +, , ◊ , ο ) and correspondingcurves simulation (⎯) in the MODEL 3.1.2 and MODEL 3.2.2, with parameter valuesfrom Table 2 and Table 4.

AnalysisAnalysisAnalysisAnalysis ofofofof thethethethe biologicalbiologicalbiologicalbiological switchigswitchigswitchigswitchig

The biological switching effect was illustrated by analysis of the semi-empiricalMODEL 5.1 (Eqs. 30-32), which generated from growth model (Eq. 5).

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2 1m SX P

X X XS m

M SIS

cdc cr c m ccdt PK cK

µ ⎛ ⎞⋅= = ⋅ ⋅ − − ⋅⎜ ⎟

⎝ ⎠+ +(30)

2 1S ms S P

S XS mm

M SIS

dc v c cr ccdt PK cK

⎛ ⎞⋅− = − = ⋅ ⋅ −⎜ ⎟

⎝ ⎠+ +(31)

2 1mp SP P

P XS mm

M SIS

v cdc cr ccdt PK cK

⋅ ⎛ ⎞= = ⋅ ⋅ −⎜ ⎟

⎝ ⎠+ +(32)

The model parameters mP and mmP represent the corresponding ethanol concentration,which stopped cell growth and ethanol synthesis respectivelly. They are determined inthe independent experiment.

In the Figure 3a, growth cells kinetic simulated in the Model 5.1 when the parametervalues are m mmP P= =84 g/dm3 was presented. The process simulation with differentparameters values mP =84 g/dm3 and mmP =95 g/dm3 was presented in the Figure 3b.The Figure 3 and corresponding Fischer’s test show that the models with differentparameter model values are more valid, especially in the domain of highly ethanolconcentration. Therefore, at kinetic simulations by deterministic models, thebiologically limiting was included in the mathematical model.

Figure 3. Cell growth kinetic simulated in the MODEL 5.1 with parameter valuesm mmP P= =84 g/dm3 (a) and with different parameter values mP =84 g/dm3 and

mmP =95 g/dm3 (b). The experimental data at different initial glucose concentration 50,100, 150, 200 and 250 g/dm3 (x, +, , ◊, ο) and corresponding simulation curves (⎯).

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CONCLUSIONCONCLUSIONCONCLUSIONCONCLUSION

In this paper, the base of the bio kinetic model for free and immobilized cells wasgenerated. The structuring of kinetic model in the form of the differential equationsystem, which is simultaneously solved at simulation, enables bioprocess kineticanalysis and modeling including different mechanisms of product synthesis. Thekinetic model structure consists of the model of biomass kinetic growth and kineticmodels of substrate consumption and product synthesis.Four models are mechanismically based on the mass and energy balances, dynamicequilibrium, biochemical reaction rate, conservation of atomic species andstoichiometric of reaction. Other kinetic models are mixed semi-empirical, derived bymodification with empirical terms that were determined in independent experiments.The interaction of biological and biochemical process is analyzed for growth-associated and non-growth-associated mechanisms of ethanol biosynthesis fromglucose. It is confirmed that the method of model structure modification withappropriate validation, can be successfully applied for confirmation of bioprocessmechanism, choice of the convenient or elimination inadequate.From generic kinetic models, the biomass growth model taking into account non-competitive substrate and product inhibition, and non-growth pattern product forming,was confirmed. The validations of the semi-empirical model were confirmed, but moreof their terms are not generated basing on physical and chemical laws or theirbiological switch, which are necessary to be researches.

NomenclatureNomenclatureNomenclatureNomenclature

Xc gX dm-3 concentration of cells (dry weight)Sc gS dm-3 concentration of substrate

Pc gP dm-3 concentration of producttot

calcF - Fisher’s distribution for validation testX

calcF ,S

calcF ,P

calcF Fisher’s distribution for partial validation test

MK gS dm-3 Monod’s constant of cells growth rateNK gS dm-3 Monod’s constant of cells growth rate related to nitrogen

IPK gP dm-3 inhibition constant for competitive inhibition by productISK gS dm-3 inhibition constant for noncompetitive inhibition by substrate

m gX gX-1 h-1 maintenance coefficient for cells growthmP gS dm-3 critical product concentration above reaction cannot proceed

mS gS dm-3 critical substrate concentration above reaction cannot proceed

Xr gX dm-3 h-1 rate of growth (production) of cells related to glucoseSr gS dm-3 h-1 rate of substrate consumption

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Pr gP dm-3 h-1 rate of product formationNr gX dm-3 h-1 rate of growth (production) of cells related to nitrogen

corr - correlation coefficient

X / NY gX gN-1 coefficients of cells yield related to nitrogen

GreekGreekGreekGreek symbolssymbolssymbolssymbols

µ gX gX-1 h-1 specific growth rate

mµ gX gX-1 h-1 maximum specific growth rate

sν gS gX-1 h-1 specific rate of substrate consumption

Pν gP gX-1 h-1 specific rate of product formation

mpν gP gX-1 h-1 maximum specific rate of product formation

msν gS gX-1 h-1 maximum specific rate of substrate consumption2adσ - nonadequacy variance

)Y( exp2σ - variance of the experimental data

REFERENCESREFERENCESREFERENCESREFERENCES

[1] T.F. Edgard, D.M. Himmelblau, Optimization of chemical processes, McGraw-Hill, New York, 1988.

[2] J.A. Roels, N.W.F.Kossen, Progress in Ind. Microbial, 14141414 (1978).[3] O. Levenspiel, Biotechnol. Bioeng., (1980) 1671.[4] L. Michaelis, M.L. Menten, Biochem. Z., 49494949, (1913) 333.[5] J. Monod, Hermann et Cie, Paris, 74 (1958).[6] J. Monod, Ann. Inst. Pasteur, Paris, 79797979 (1950) 390.[7] J. Monod, Ann Rev Microbiol, 3333, (1949) 371.[8] F. Jacob, J. Monod, J. Mol. Biol., 3333, (1961) 318.[9] Y. Liu, Biochemical engineering journal 32323232, (2006) 102.[10] L. von Bertalanffy, General system theory, Penguin books Ltd, 1968.[11] S. Kuzmanova, S. Zerajic, F. Popovska, E. Vandeska, Kem. Ind., 39393939 9, (1990) 421.[12] S. Kuzmanova, S. Zerajic, E. Vandeska, Mikrobiologija, 30303030, 2 (1993) 77.[13] D. Voet, J.G. Voet, Biochemistry, John Wiley & Sons, New York, 1990.[14] C.G. Sinclair, D. Cantero, Fermentation modelling. In: Fermentation a practical

approach. IRL PRESS at Oxford University Press, New York, 1990, p. 92.[15] M.C. Bezenger, J.M. Navarro, Biotechnol. Bioeng., 31313131, (1988) 747.[16] M. Saita and J.C. Slaughter, Enzyme Microbiol. Technol., 6666, 375, 1984.[17] R. Luedeking, E.L. Piret, 1111 (1959) 393.[18] K. Levenberg, Q. Appl. Math., 2222 (1944) 164.[19] D. Marquardt, S.I.A.M. J. Appl. Math., 11111111 (1963) 431.

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[20] H.H. Rosenbrock, Comput. J., 3333 (1960) 175–184.IzvodIzvodIzvodIzvod

KINETIKINETIKINETIKINETIČČČČKIKIKIKI MODELIMODELIMODELIMODELI BIOPROCESABIOPROCESABIOPROCESABIOPROCESA SASASASA SLOBODNIMSLOBODNIMSLOBODNIMSLOBODNIM IIII IMOBILISANIMIMOBILISANIMIMOBILISANIMIMOBILISANIMĆĆĆĆELIJAMAELIJAMAELIJAMAELIJAMANauNauNauNauččččnininini radradradradStankoStankoStankoStanko ŽŽŽŽerajierajierajierajićććć1111,,,, JelenkaJelenkaJelenkaJelenka SavkoviSavkoviSavkoviSavkovićććć-Stevanovi-Stevanovi-Stevanovi-Stevanovićććć22221Tehnološki fakultet, Leskovac, Srbija2Tehnološko-metalurški fakultet, Beograd, Srbija

Kinetički modeli su osnovna znajna o biohemijskim i mikrobiološkim procesima.Predmet rada je istraživanje kinetičkih modela zasnovanih na biohemijskim imikrobiološkim mehanizmima rasta ćelija, nestajanja supstrata i formiranja proizvoda.Inženjerski cilj je proces biosinteze etanola kao goriva iz obnovljivih izvora.Inženjerski cilj zahteva uključivanje inhibitornih efekata visoke koncentracije supstratai etanola u kinetičke modele. Istražene su interakcije mehanizma formiranja etanola sarastom biomase. Modeli mehanizama formiranja etanola ugrađeni su u složenustrukturu kinetičkih modela. Istraživanja kinetičkih modela izvedena su računarskimsimulacijama i eksperimentalnim verifikacijama na procesu biosinteze etanola izglukoze sa slobodnim i imobillisanim ćelijama S. cerevisiae. Ekperimentalnomvalidizacijom ocenjeni su parametri modela i ograničenja. Kreirana je bazaverifikovanih determinističkih i poluempirijskih kinetičih modela. Baza modela semože dalje koristiti u izgradnji procesnih simulacionih modela.

KljuKljuKljuKljuččččnenenene rerererečččči:i:i:i: Kinetički modeli, biosinteya etanola, sobodne i imobilisane ćelije,struktura i validizacija modela.

THEKINETICMODELSOFTHEBIOPROCESSWITHFREEANDStankoŽerajić1,JelenkaSavković-Stevanović2Keywords:Kineticmodel,ethanolbiosynthesis,frINTRODUCTIONFreesuspendedandimmobilizedcellsMATERIALSANDMETHODS

MODELSDEVELOPEMENTThestructureofkineticmodelforbioethanolsyntThemodelsofbiomassgrowthkineticsKineticmodelsofthesubstrateconsumptionKineticmodelsoftheproductsynthesisGrowthassociatedmodelproductformationNon-GrowthassociatedmodelproductformationParameterestimationandmodelvalidation

RESULTSANDDISCUSIONCONCLUSIONNomenclatureGreeksymbolsREFERENCESIzvod

KINETIČKIMODELIBIOPROCESASASLOBODNIMIIMOBILINaučnirad

StankoŽerajić1,JelenkaSavković-Stevanović2