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Bioautomation, 2007, 7, 34 45 ISSN 1312 451X

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Mathematical Description of Functional Statesin E. coli Fed-batch Cultivation Processes

Olympia Roeva*1

, Kalin Kosev2

, Stoyan Tzonkov1

1Centre of Biomedical Engineering Prof. Ivan Daskalov Bulgarian Academy of Sciences105 Acad. G. Bonchev Str., Sofia 1113, Bulgaria

E-mail: {olympia, tzonkov}@clbme.bas.bg

2University of Perugia Piazza University, 1 Perugia 06123, Italy E-mail: [email protected]

*Corresponding author

Received: February 16, 2007 Accepted: September 27, 2007

Published: October 24, 2007

Abstract: This paper presents an overview of a biochemical correspondence to defined functional states based on specific metabolic mechanisms. For Escherichia coli fed-batchcultivation processes a set of four functional states is considered. It is shown that the processcan be divided into several functional states by considering the cell metabolism in moredetail. As a result through the proposed functional states the changes in metabolic pathwayscan be described accurately. For each functional state a respective local model is proposed.Simulations of the E. coli cultivation process with functional state modelling are presented.

Furthermore different functional states in a real E. coli MC4110 fed-batch cultivation process are identified and local models are developed. By simulations and comparing resultsto experimental data is shown that the concept of functional state modelling works in

practice and leads to more precise and adequate mathematical description.

Keywords: Functional states, Metabolism, Mathematical description, Fed-batch processes, Escherichia coli.

IntroductionThe mathematical description of a cultivation process on the basis of a number of separatefunctional states has been shown useful in modelling and control of such complex processes[8, 15, 23, 24]. In each functional state (top level of the system hierarchy) the process isdescribed by a conventional type of model, called local model, which is valid only in thisfunctional state. At the second hierarchical level some numeric detection algorithms and/orrules based on expert knowledge can be used for the recognition of the functional state andstate transitions. A set of local models together with functional state dynamics can be usedto describe, monitor and control the overall growth process.

Murray-Smith and Johansen [12] have provided an introduction to the functional state conceptand have illustrated it with a simple wire model. The methods for detection of changes inprocess dynamics and transitions between states are applied to a practical biotechnical processcontrol task. One of the most important characteristics of biotechnological processes, whichmake the control design more difficult, is the change of cell population state. In most casesthis change is expected in view of the fact that the cells pass through different phases andgrowth states. In many batch-type cultivation processes, the functional states would naturallybe identified with the different phases of the process lag phase, log phase, stationary phaseand death phase. In a fed-batch or continuous process the main problems are how to divide the

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where S is concentration of substrate (glucose), [g/l]; pO 2 concentration of dissolvedoxygen, [%]; A concentration of acetate, [g/l]; crit S and 2crit pO critical values of substrateand dissolved oxygen concentrations.

The growth of E. coli in this functional state could be represented through the reaction:1

1 8 2 5k S k pO X k A

+ + , (4)

where X is biomass concentration, [g/l]; k 1, k 5 and k 8 are reaction coefficients, [-] and 1 isreaction rate, [h -1].

In this state the main carbon source is the glucose and the acetate excretion is the result of ametabolic overflow mechanism [21]. Overflow metabolism has been attributed to anenzymatic limitation in the TCA cycle. If the rate of dissolved oxygen utilization issufficiently high, the reduced cofactors generated by glucose consumption are reoxidized inthe electron transport chain, which serves the dual purpose of maintaining an optimal redoxenvironment and generating energy by oxidative phosphorylation. If the rate of glucoseconsumption is greater than the capacity to re-oxidize the reduced equivalents, metabolicintermediates accumulate to maintain the redox balance [27].

Mixed oxidative state (FS II) The process enters this state when the following conditions are available:

2 2,crit crit S S pO pO and 0 A > . (5)

Here both sugar and produced acetate are co-metabolized through the oxidative pathways inthe state. The bacteria E. coli are able to re-oxidize the reduced metabolic intermediates thatwere accumulated during the first functional state. In this functional state accumulated acetateis also metabolized. Therefore the growth process could be described through the next tworeactions:

2

2

2 9 2

6 10 2 ,

S

A

k S k pO X

k A k pO X

+

+ (6)

where k 2, k 6, k 9 and k 10 are reaction coefficients, [-] and 2 S , 2 A are reaction rates, [h-1].

Complete sugar oxidative state (FS III) The rules for recognition of FS III have to meet the requirements:

2 2,crit crit S S pO pO and 0 A = . (7)

In this state, sugar is completely oxidized to water and carbon dioxide. There is sufficientquantity of dissolved oxygen in the media which helps with the reoxidizing of the metabolicintermediates in the cell (NADH and FADH). This means that there will be no acetateformation and the pyruvate will enter directly the TCA cycle and will be completely oxidizedto water, carbon dioxide and several molecules of NADH and FADH. In this case thefollowing reaction is used:

3

3 11 2k S k pO X

+ , (8)

where k 3 and k 11 are reaction coefficients, [-] and 3 is rate of the reaction, [h-1].

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Second acetate production state (FS V)The conditions for this state are:

2 2,crit crit S S pO pO < and 0 A > . (9)

When the dissolved oxygen becomes the limiting factor for E. coli growth, acetate isproduced. Due to the low level of dissolved oxygen metabolic intermediates can not bereoxidized. This means that the process of glucose metabolism will stop at the stage of acetyl-coA formation by the pyruvate dehydrogenase complex. And from there the acetyl-coA willbe oxidized to acetate which will be stored or excreted from the cell. This acetate will then beused for consumption by the cell if the dissolved oxygen concentration exceeds the criticalminimum. The growth of E. coli in FS V could be interpreted through the following reaction:

5

4 7k S X k A

+ , (10)

where k 4 and k 7 are reaction coefficients, [-] and 5 is reaction rate, [h-1].

General state space dynamical local models

The mathematical model of E. coli fed-batch cultivation processes can be presented by thefollowing four submodels (local models) according to which functional state the process is:

Local model for first acetate production state (FS I):

( )

( )

1

1

1

1 1

1

1 1

1

2 1 1

S

S

S

22 1 2 2

S

1

1

1

,

inmax

inmax in

S / X

inmax

A / X

*inmax L

pO / X

in

F dX S X X

dt k S V

F dS S X S S dt Y k S V

F dA S X A

dt Y k S V

F dpO S X pO k a pO pO

dt Y k S V

dV F

dt

= +

= + +

= +

= + +

=

(11)

where 11

1S

maxS

k S = + ,1

1

1

S / X k Y = ,

1

5

1

A / X k Y = ,

2 1

8

1

pO / X k Y = and ( )1 1 2 2

* LOTR k a pO pO= .

In this functional state the specific growth rate is described using Monod kinetics. Thespecific rate of sugar consumption, the specific acetate rate and the specific dissolved oxygenconsumption rate are directly proportional to the specific growth rate, i.e. to the Monodkinetics.

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Local model for mixed oxidative state (FS II):

( )

( )

2

2

2

2 2

2

2

2 2 2 2 2

S A

S

A

22

S A

2 2 2

1

1

1 1( )

,

inmax max A

inmax inS / X

inmax A

A / X

inmax max A

pO / S pO / A

* L

in

F dX S A X X X

dt k S k A V

F dS S X S S dt Y k S V

F dA A X A

dt Y k A V

F dpO S A X pO

dt Y k S Y k A V

k a pO pO

dV F

dt

= + + +

= + +

= +

= + ++ +

+

=

(12)

where2

2

2S

S maxS

k S = + , 2

A A max A

Ak A

= + ,2

2

1

S / X

k Y

= ,2

6

1

A / X

k Y

= ,2 2

9

1

pO / S

k Y

= ,

2 2

10

1

pO / A

k Y

= and ( )2 2 2 2* LOTR k a pO pO= .

In mixed oxidative state the specific growth rate is expressed according to the sum of twoterms. The first term is describing the contribution of glucose and the second thecontribution of acetate to bacterial growth. Both terms have the structure of Monod kinetics.A Monod model is also used for the specific acetate and glucose rates, respectively, accordingto acetate concentrations and substrate concentrations. The specific dissolved oxygenconsumption rate is obtained as a sum of two terms, which are directly proportional to thespecific glucose rate and specific acetate rate, respectively.

Local model for complete sugar oxidative state (FS III):

( )

( )

3

3

3

3 3

3

2 3 3

S

S

22 3 2 2

S

1

0

1

,

inmax

inmax in

S / X

in

*inmax L

pO / X

in

F dX S X X

dt k S V

F dS S X S S

dt Y k S V F dA

X Adt V

F dpO S X pO k a pO pO

dt Y k S V

dV F

dt

= +

= +

+=

= + +

=

(13)

where3

3

3S

max

S

k S =

+,

3

3

1

S / X

k Y

= ,2 3

11

1

pO / X

k Y

= and ( )3 3 2 2* LOTR k a pO pO= .

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In the complete sugar oxidative state the specific rates of bacterial growth, sugar consumptionand dissolved oxygen consumption are described by Monod models. Here the specific acetateproduction rate is zero.

Local model for second acetate production state (FS V):

( )

( )

5

5

5

5 5

5

5 5

S

S

S

22 5 2 2

1

1

0

,

inmax

inmax in

S / X

inmax

A / X

*in L

in

F dX S X X

dt k S V

F dS S X S S

dt Y k S V

F dA S X A

dt Y k S V

F dpO X pO k a pO pO

dt V dV

F dt

= +

= + +

= +

= +

=

(14)

where5

5

5S

max

S k S

= + ,5

4

1

S / X

k Y

= ,5

7

1

A / X

k Y

= and ( )5 5 2 2* LOTR k a pO pO=

In the second acetate production state the specific rates of bacterial growth and acetateproduction depend on the glucose concentration. The specific rate of sugar consumption isdescribed again by Monod model. Here the specific dissolved oxygen consumption rate iszero.

Simulation of the E. coli fed-batch cultivation process with functional statemodellingProposed local models are used for numerical simulations of the functional states in the

E. coli fed-batch cultivation process. For process simulation the ode45 integration algorithm is used under Matlab 5.3 environment [26].

Functional state transitions are determined according to the conditions on the concentrationsof substrate, dissolved oxygen and acetate. For considered fed-batch cultivation process it isnecessary to maintain the residual substrate concentration at a very low level. A low residual

level of substrate is advantageous in: (i) removing repressive effects of rapidly utilized carbonsources and maintaining conditions in the culture within the aeration capacity of thebioreactor; (ii) avoiding the toxic effects of a medium component. According to the abovementioned and based on the specific peculiarity of the E. coli cultivation process the value forcritical substrate concentration S crit = 0.02 g/l is assumed. The critical dissolved oxygenconcentration is pO 2crit = 21%.

When the functional state is changed the local models are changed correspondingly. Theinitial values for simulations in the new functional state are the last simulated values in theprevious functional state so that the trajectories became continuous. The parameter valuesused in the simulation are presented in Table 1. The parameter values are chosen based on thetypical values published in the literature regarding E. coli fed-batch cultivation process [2, 4,9, 10, 14].

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Table 1. Parameter values used in the simulation

FS I FS II FS III FS V Parameter Value Parameter Value Parameter Value Parameter Value

1max , [h-1] 0.32 2max , [h

-1] 0.30 3max , [h-1] 0.50 5max , [h

-1] 0.35

1Sk , [g/l] 0.02 max A , [h -1] 0.15 3Sk , [g/l] 0.03 5Sk , [g/l] 0.03

1S / X Y , [-] 0.46 2Sk , [g/l] 0.03 3S / X Y , [-] 0.50 5S / X Y , [-] 0.50

1 A / X Y , [-] 0.10 Ak , [g/l] 0.04 2 3 pO / X Y , [-] 0.50 5 A / X Y , [-] 0.10

2 1 pO / X Y , [-] 0.50 2S / X Y , [-] 0.49 3 Lk a , [h

-1] 80.0 5 Lk a , [h-1] 30.0

1 Lk a , [h-1] 52.0 2 A / X Y , [-] 0.09

2 2 pO / S Y , [-] 0.20

2 2 pO / AY , [-] 0.40

2 Lk a , [h -1] 87.0

Based on series of simulations a substrate feed rate profile (Fig. 1) is designed to ensure thepresence of all considered functional states in the simulation. In practice it is possible acertain functional state to appear more than once and some of them do not appear at all. In thiscase the aim is to present the dynamics of a cultivation process during all defined functionalstates. The process simulation is presented in Fig. 2.

Fig. 1 Feed rate profile of E. coli fed-batch cultivation process

As it is presented in Fig. 2 the defined functional state ( FS I , FS II , FS III and FS V ) can beclearly recognized. In the first acetate production state the sugar concentration decreases veryrapidly and acetate is accumulated in large quantities. The production of acetate continuesuntil the end of the first acetate production state. In the mixed oxidative state bacteria begin toutilize acetate together with sugar. At the end of this state the acetate concentration is almostzero and the sugar concentration is below the critical sugar concentration. In the next state

(complete sugar oxidative state ) there is no acetate production, the bacteria uses only sugar asmain energy source. In all above-mentioned functional states there is sufficient dissolved

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5x 10 -3 E. coli fed-batch cultivation process - Feed rate profile

Time [h]

FS I FS II FS III FS V

Feed rate [l/h]

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In this fed-batch cultivation process only two functional states appear since there is sufficientdissolved oxygen during the first four hours. In the beginning of the cultivation the dissolvedoxygen and the glucose concentrations are above the corresponding critical levels. Theprocess is in the first acetate production state ( FS I ). At the 8.7h of the cultivation process theconcentration of dissolved oxygen becomes below the dissolved oxygen critical level and theprocess enters in the second acetate production state ( FS V ). Here the concentration of thesugar concentration is below the critical sugar concentration and the acetate productioncontinues.

The simulation results show that the developed local models based on functional states predictsuccessfully the variation of glucose consumption, biomass concentration, acetate formationand dissolved oxygen consumption during the fed-batch fermentation of E. coli MC4110 . Theresults illustrate that the concept of functional states leads to the precise and adequatemathematical description.

ConclusionIn this paper based on an overview of specific metabolic mechanisms in E. coli fed-batchcultivation processes a set of four functional states is considered. By simulations it is shownthat the process can be divided into several functional states by considering the cellmetabolism in more detail. For each functional state a corresponding local model isdeveloped. The functional state modelling is applied to a real E. coli MC4110 fed-batchcultivation process. As a result different functional states are identified and local models areproposed. The results and the simulations indicate that the process can be rather well modeledwith the functional state approach, show a good efficiency and the ability of the appliedapproach to use in practice.

The functional state concept could be applied in modelling of E. coli cultivation processes asan alternative for the conventional global modelling. It is difficult to develop globalmodels, which are valid for all different process phases. Although structured models areknown to describe the E. coli cultivation process well, they are quite complex to use inindustrial-scale production in contrast to local models.

AcknowledgementsThis work is partially supported from National Science Fund Project MI 1505/2005.

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