microbial growth on multiple substrates

8/14/2019 Microbial Growth on Multiple Substrates

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Microbial Growth on MultipleSubstrates

Professor Dhinakar S. Kompala

Department of Chemical and BiologicalEngineeringni!ersit" of Colorado

Boulder# Colorado $%&%'

(ith Pol"math programs written b" ).S.*ogler and Som Ghosh

Modifications to Monod Model

The empirically based Monod growth rate equation

has become popular, compared to theother proposed rate equations for cell growth, dueto its similarity with the mechanistically derived

Michaelis-Menten rate law forenzyme-catalyzed reaction rate. The Monodequation is capable of explaining or simulating the

exponential growth phase followed by thedecelerating growth phase in the cell concentrationduring batch growth dynamics, when coupled withdynamic balance equation for the substrateconcentration:

where YC/Sis the stoichiometric yield coefficient ofgrams of cell mass produced from a gram ofsubstrate.


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igure !. Typical growth phases in batch cultures of

bacterial cells

The initial lag and accelerating growth phases whichare not simulated by the Monod growth rateequation can be easily simulated by ma"ing a slightmodification to the original Monod equation givenabove. This modification invo"es a rate-limitingenzyme involved in the growth processes that maynot be present at sufficient levels in the inoculum orstarting culture. #hile cell growth is a complex

process mediated by thousandsof enzymes, it maybe sufficient to hypothesize $ust a single enzymethat may be rate-limiting during the initial laggrowth phase for the purpose of simulating the lag.

%ncorporating the effect of varying "ey enzymeconcentration into the Monod growth "inetics, wecan write

&'.(.).*+

where ERrepresents the relative amount of the "eyenzyme in the cell. This modified Monod rateequation follows the Michaelis-Menten rate equationclosely, which is more correctly written as


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&'.(.).)+

where ETrepresents the total enzyme concentration

in the reaction mixture.

'elative "ey enzyme content inside the cells, ER,

may be written as , where eis theintracellular content of the "ey enzyme, with theunits of g enzymeg cell mass, and emaxis is themaximum enzyme concentration in the cell, alsowith units of g enzymeg cell mass. The balanceequation for the intracellular enzyme content can be

written in terms of eCc,which has the units of genzymeculture volume, as

&'.(.).+

where aand bare enzyme synthesis anddegradation rate constants. sing the product rule,we can expand the above equation to write thebalance equation for eas:

&'.(.)./+

The last term in the above balance equation for theintracellular enzyme is the dilution term due to cellgrowth, which will be obtained similarly for allintracellular species. This equation can be solved toproduce the curve added at the bottom of the batchgrowth dynamics as shown in igure 0. Themaximum level of the intracellular enzyme emaxcanbe determined easily by setting the above equationto zero and solving for its steady state value. %nterms of the model parameters, the maximum levelof intracellular enzyme obtained during theexponential growth phase &when Csis much greaterthan Ks+ can be derived as


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&'(.).(+

1tarting with a low or zero content of the "eyenzyme needed for growth on a given substrate&representative of poor quality starter culture orinoculum+, the presence of the substrate inducesthe synthesis of the "ey enzyme. The low or zeroenzyme content causes the initial lag phase of nocell growth while increasing enzyme content resultsin the accelerating growth phase. The enzymeconcentration reaches and stays at its maximum,emax, during the exponential growth phase, whichshows up as a linear increase in the logarithmic plot

of cell concentration. The slope of this line on thesemi-logarithmic plot below is the maximumspecific growth rate, mmax .2s the substrate getsdepleted, the growth rate or the slope of this curvedecelerates and becomes zero when the substrateis completely consumed. #ith the substrate nolonger present, the "ey enzyme synthesis stops andenzyme degradation during the stationary anddeath phases reduces the intracellular enzymecontent to low levels.

P+,-M/) code0

Calculated !alues of the DE1 !ariables

3ariable initial value minimal value maximalvalue final value

t 4 4 ) )

5c 4.! 4.! !.( !.(

5s ) 6.6*7-!! ) 6.6*7-!!

e 6.*7-4 6.*7-4 !.47-4)!.407-4)

8cs 4.) 4.) 4.) 4.)


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mumax ? 7'?5s&s @ 5s+

Figure 2. Dynamic profiles of substrate concentration

and intracellular enzyme content along with thelogarithm of cell mass concentration.

5lic" here for the Aolymath 5ode

7.4.&.8 Se9uential Growth :

Diau;ie

5onsider the cell growth medium in a batchbioreactor that contains two different carbonsubstrates, 1!and 10,which are each capable ofsupporting cell growth. or the growth on thesubstrate !:

5ells @ 1ubstrate ! &e.g. glucose+ B more 5ells @products,

the modified Monod growth rate expression is

&'.(.).6+

or growth on the substrate 0:
http://www.engin.umich.edu/dept/che/OldFiles/OldFiles/kinetics/07chap/html/singlesub.polhttp://www.engin.umich.edu/dept/che/OldFiles/OldFiles/kinetics/07chap/html/singlesub.pol


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5ells @ 1ubstrate 0 &e.g. lactose+ B more 5ells @products,

the modified Monod growth rate expression is

&'.(.).9+

where m1and m2are the specific cell growth rateson individual substrates 1!and 10respectively.

#hen both the substrates are present in a batchbioreactor, microbial cells do not consume bothsubstrates simultaneouslyor additivelybut insteadgrow on the sugars sequentially. That is, the

maximum specific growth rate on mixed substrates:

.

The sequential consumption results in an interestingpattern of the cells first consuming one preferredsubstrate and then after an intermediate lag phase,consuming the remaining less preferred substrate.That is, the less preferred substrate is not utilizeduntil the preferred substrate is completelyconsumed and is no longer available in the growthmedium. This sequential utilization of twosubstrates in batch cultures has been observed innumerous experiments by Monod, who has termedthis phenomenon as the CdiauxieC &Dree" for twogrowth phases+.

The consistent characteristic of the diauxic growthphenomenon is that the preferred substrateprovides the faster growth rate, i.e. mmax,!E mmax0.The emzyme growth curve along with the cell andsubstrate concentration are shown in igure *.


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Figure 3. Diauxic growth of bacteria on twosubstrates, showing also the intracellular content of the

two hypothetical key enzymes for consuming eachsubstrate.

The diauxic growth is observed in most or all well-documented batch cultures on multiple substrates.The preference for faster growth rate in the firstgrowth phase has been suggested to be aconsequence of evolutionary pressures on themicrobes to grow at the fastest growth rate possible


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&'.(.).!4+

&'.(.).!!+

&'.(.).!0+

for i> !,0 &'.(.).!*+

where

&'.(.).!)+

e.g. and

The last term in the equation &'.(.).!*+ is thetypical dilution of an intracellular species due to cellgrowth. ;uring the exponential growth on highconcentration of a single substrate, the maximumlevel of the intracellular "ey enzyme can be shownas

&'.(.).!+

#ith these CcyberneticC model equations, thediauxic growth phases shown above have beensimulated with typical model parameters values.


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The model parameters are the Monod growthparameters for growth on each single substrate&which can be determined experimentally+ andadditional model parameters aiand bifor thesynthesis and degradation of the two "ey enzymes.2s the rate-limiting "ey enzymes are hypothetical,

even though they may be identifiable in many cases&such as, b-galactosidase for the utilization oflactose+, these additional model parameters aregiven some representative values, rather thanexperimentally measured.

E;ample 2.7.4.8.80 Diau;icGrowth0 Ma;imi 4.9 hr-!, mmax,0>4./ hr-!, !> 4.! gF, 0> 4. gF, ai> 4.444! hr-!, bi> 4.4 hr-!+ and initial conditions for the two"ey enzymes &e!,4at closer to its maximum value of6.* x !4-and e0,4at a lower value of ! x !4-/+, thegrowth rate on substrate ! m!remains higher thanm0until substrate ! is completely consumed.5onsequently cybernetic variables u1and v1for thefirst enzymeHs synthesis and activity remain close totheir maximum value of ! during the first growth

phase.

P+,-M/) code0 32un this program withS/=** +DE algorithm6



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t 4 4 !4 !4

5c 4.! 4.! 9.( 9.(

5s! ) 0.)4/7-0 )0.)4/7-0

5s0 04 -0.4(07-!! 04*.!*7-0

e! 6.*7-4 !.*0*7-4 !.4)97-4)!.*0*7-4

e0 !.47-4/ 0.*697-4( !.!7-4)!.**7-4)

mu!max 4.9 4.9 4.94.9

mu0max 4./ 4./ 4./4./

! 4.! 4.! 4.! 4.!

0 4. 4. 4. 4.

beta! 4.4 4.4 4.44.4

beta0 4.4 4.4 4.44.4

alpha! !.47-4) !.47-4) !.47-4)!.47-4)

alpha0 !.47-4) !.47-4) !.47-4)!.47-4)

e!max !.4*7-4) !.4*7-4) !.4*7-4) !.4*7-4)

e0max !.*67-4) !.*67-4) !.*67-4) !.*67-4)


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7! 4.(66 4.!06* 4.99/(*)04.!06*

70 4.44/ 4.44!06 4.96!0*9!4.664/(

mu! 4./90*)! /.!*/7-04.6/*!(0! /.!*/7-0

mu0 4.44*64)9 -0.**97-!!4.)()!)0 /.90*7-0

u! 4.99)*)) -!(.(4090!4.996(69/ 4.)/96((9

u0 4.44)// 4.44!0!4)!6.(4090! 4.*4!00!

mumax 4./90*)! /.90*7-04.6/*!(0! /.90*7-0

v! ! ).4967-4) !4.66/*(6

8cs! 4.) 4.) 4.) 4.)

8cs0 4.) 4.) 4.) 4.)

v0 4.44)9( -!.4/)6(9 !!

+DE 2eport 3S/=**6

;ifferential equations as entered by the user

&mu!?v!@mu0?v0+?5c

-mu!?v!?5c8cs!

-mu0?v0?5c8cs0

alpha!?5s!&!@5s!+?u! -beta!?e! - &mu!?v!@mu0?v0+?e!


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alpha0?5s0&0@5s0+?u0 -beta0?e0 - &mu!?v!@mu0?v0+?e0

7xplicit equations as entered by the user

.9

./

.!

.

.4

.4

.444!

.444!

alpha!& mu!max@ beta!+

alpha0& mu0max@ beta0+

e!e!max

e0e0max

mu!max?7!?5s!&!@5s!+

mu0max?70?5s0&0@5s0+

mu!&mu!@mu0+

mu0&mu!@mu0+

if &mu!Emu0+ then &mu!+ else&mu0+

mu! mumax

.)

.)

mu0mumax


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Figure E.!.".#$ %imulations of the cybernetic model,

showing the profiles for the two substrateconcentrations &%# and %2 on the left axis' and

logarithm of cell mass &ln ( on the right axis' during a

typical diauxic growth.

The intracellular enzyme levels for the samesimulation are shown below to highlight the role ofthe cybernetic variable u1and u2in the synthesis ofe1and e2. These are plotted using the last program,by plotting e!,e!, u! and u0 vs time.

P+,-M/) /B,E

e! 6.*7-4 !.*0*7-4 !.4)97-4)!.*0*7-4
http://www.engin.umich.edu/dept/che/OldFiles/OldFiles/kinetics/07chap/html/twosubstrate.polhttp://www.engin.umich.edu/dept/che/OldFiles/OldFiles/kinetics/07chap/html/twosubstrate.pol


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e0 !.47-4/ 0.*697-4( !.!7-4)!.**7-4)

u! 4.99)*)) -!(.(4090!4.996(69/ 4.)/96((9

u0 4.44)// 4.44!0!4)!6.(4090! 4.*4!00!

Figure E.!.".2$ %imulations of the cybernetic model

showing the profiles of the two cybernetic )ariables

&u#and u2on the left axis' and two enzymes &e#and e2on the right axis'

;uring the first growth phase, the cyberneticvariable u1ta"es values close to unity, indicatingpreferential synthesis of enzyme ! and repression&suppression of synthesis+ of enzyme 0 as u2isclose to zero. 2fter the first substrate is mostlyconsumed, the growth rate m1on that substratefalls to zero, triggering the switch in the cyberneticvariables and inducing synthesis of enzyme 0.


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E;ample 2.7.4.8.>. 0 Diau;icGrowth0 Effect of Preculturing inSubstrate >

%t may be suspected from the above simulation thatpreferential utilization of substrate ! is due to thehigh level of enzyme ! assumed as its initial value.This high initial value for e1is chosen to indicatepreculturing the inoculum in substrate !. %f theinoculum is precultured in substrate 0, the initialvalue for enzyme 0 should be higher and that forenzyme ! should be assumed much lower.

The two figures below show simulation results withthe altered initial conditions for the two enzymes,while "eeping all other initial values and modelparameter values identical to those in the aboveexample. The diauxic lag gets significantlyshortened, along with significant consumption of 10during the first growth phase. Ievertheless,substrate ! is gradually preferred with increasingculture time and is completely consumed during thefirst growth phase.

P+,-M/) code0



t 4 4 !4 !4

5c 4.! 4.! .( .(

5s! ) !.((7-! )!.((7-!

5s0 !4 !.)!(7-!) !4!.)!(7-!)


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e! !.47-4/ !.47-4/ /.)!07-4*.49/7-4

e0 !.47-4) (.06)7-4 !.)!7-4)!.!/7-4)

mu!max 4.9 4.9 4.94.9

mu0max 4./ 4./ 4./4./

! 4.! 4.! 4.! 4.!

0 4. 4. 4. 4.

beta! 4.4 4.4 4.44.4

beta0 4.4 4.4 4.44.4

alpha! !.47-4) !.47-4) !.47-4)!.47-4)

alpha0 !.47-4) !.47-4) !.47-4)!.47-4)

e!max !.4*7-4) !.4*7-4) !.4*7-4) !.4*7-4)

e0max !.*67-4) !.*67-4) !.*67-4) !.*67-4)

7! 4.449 4.449 4./49!(9/4.09)*9)

70 4./ 4.)(*)*! 4.9!/(9(4.()6

mu! 4.446*)! /.0)67-!4.)(60*/6 /.0)67-!

mu0 4.*(!)06/ !./*/7-!)4.00(49 !./*/7-!)

u! 4.40!9/) 4.4!*)/!*4./*(09 4.0(/*(9*


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u0 4.9(64* 4.*/)0(!4.96/*6( 4.(0*/04(

mumax 4.*(!)06/ !./*/7-!)4.00(49 !./*/7-!)

v! 4.400)(6 4.4!*/) !4.*6!9*9)

8cs! 4.) 4.) 4.) 4.)

8cs0 4.) 4.) 4.) 4.)

v0 ! 4.(099() !!

+DE 2eport 3S/=**6


&mu!?v!@mu0?v0+?5c

-mu!?v!?5c8cs!

-mu0?v0?5c8cs0

alpha!?5s!&!@5s!+?u! -beta!?e! - &mu!?v!@mu0?v0+?e!

alpha0?5s0&0@5s0+?u0 -beta0?e0 - &mu!?v!@mu0?v0+?e0


.9

./

.!

.

.4


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.4

.444!

.444!



e!e!max

e0e0max

mu!max?7!?5s!&!@5s!+

mu0max?70?5s0&0@5s0+

mu!&mu!@mu0+

mu0&mu!@mu0+


mu! mumax

.)

.)

mu0mumax


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Figure E.!.".3$ %imulations of cybernetic model with

altered initial conditions for the two enzyme le)els,with e2* e#, reflecting the preculturing of inoculum on

%2.

The gradual increase in the slope of semi log plot ofcell concentration during the first growth phase is

due to the gradual preference of substrate !, eventhough the inoculum is precultured on substrate 0and has the enzyme 0 already available for itscontinued consumption. ;uring the later parts offirst growth phase, more of the enzyme ! issynthesized &as is seen in the next simulationgraph+ resulting in the rapid consumption ofsubstrate ! and increasing growth rate &slope of thesemi log curve+. 1ignificant availability of enzyme 0at the end of first growth phase results in thereduced or non-existent diauxic lag phase beforethe second growth phase on the remainingsubstrate 0.

P+,-M/) /B,E0

e! !.47-4/ !.47-4/ /.)!07-4*.49/7-4

e0 !.47-4) (.06)7-4 !.)!7-4)!.!/7-4)


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u! 4.40!9/) 4.4!*)/!* 4./*(094.0(/*(9*

u0 4.9(64* 4.*/)0(! 4.96/*6(4.(0*/04(

Figure E.!."." %imulations of the cybernetic model

showing the two cybernetic )ariables u1and u2alongwith the profiles of intracellular enzyme contents for

the two key enzymes e1and e2. +reculturing causes the

initial )alue for e2to be much higher than that for e1.E)en with the )alues, the model predicts an increasing

preference for the substrate # during the first growth

phase.

2.7.4.> Simultaneous tili


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balance equations given above can be modified toinclude the inlet and outlet terms as below:

&'.(.).!/+

&'.(.).!(+

&'.(.).!6+

!o" i> !,0 &'.(.).!9+

where ; is the dilution rate, 51!,4and 510,4are theinlet concentrations of substrates 1!and 10respectively. %n equation '.(.).!9 for intracellularenzymes, an additional synthesis rate constant a? isincluded to ensure a low level presence of eachenzyme even in the absence of its substrate.

#ith these new dynamic balance equations forcontinuous cultures, the cybernetic model predictsthe simultaneousutilization of both substrates atsteady state for low dilution rates, as observedexperimentally. 2t increasing dilution rates, thesimultaneous utilization of both substrates changesgradually to#"e!e"entialutilization of the preferred&faster growth supporting+ substrate. 2t evenhigher &than the maximum growth rate possible in


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the chemostat: + dilution rates, washoutof cells from the chemostat is observed as alsoobserved for single substrate chemostats.

E;ample 2.7.4.>.8 Simultaneousutili 4.9 hr-!, mmax,0> 4./

hr

-!

, !> 4.! gF, 0 > 4. gF, ai> 4.444! hr

-!

, bi> 4.4 hr-!and the new parameter ai?> 4.4!a+, theabove modified cybernetic model equations ofcontinuous cultures can be simulated to plot thesteady state concentration of cell mass, substrates! and 0 over a range of dilution rates. The initialvalues for the different concentrations areimmaterial &as long as cell mass concentration isnot started at zero since there will be nospontaneous generation of life in a sterilebioreactor+ if we simulate the dynamic balanceequations &'.(.).!/-'.(.).!9+ long enough forthem to reach steady state. The inlet

concentrations of the two substrates are chosen as!4 gF each and the inlet nutrient feed is assumedto be sterile &i.e. cell mass concentration in the feedis zero+.

P+,-M/) code0 3?ote0 Sol!e this programusing the S/=** algorithm in P+,-M/)6

+DE 2eport 3S/=**6


&mu!?v!@mu0?v0+?5c - ;?5c

-mu!?v!?5c8cs! @ ;?&5s!4-5s!+


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-mu0?v0?5c8cs0 @ ;? &5s04 -5s0+

alpha!?5s!&!@5s!+?u! -beta!?e! - &mu!?v!@mu0?v0+?e! @ alphastar

alpha0?5s0&0@5s0+?u0 -beta0?e0 - &mu!?v!@mu0?v0+?e0 @ alphastar


.9

./

.!

.

.4

.4

.444!

.444!



e!e!max

e0e0max

mu!max?7!?5s!&!@5s!+

mu0max?70?5s0&0@5s0+

mu!&mu!@mu0+

mu0&mu!@mu0+


mu! mumax


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.)

.)

mu0mumax

.4!?alpha!

)

04

.)

#e run this program with ; > .! to ; > .9 withintervals of .! and collect the final values&equilibrium values+ and plot them against thecorresponding ; values. Diven are the tables for ;> .), . and ./.

Calculated !alues of the DE1 !ariables for D

@ .4


t 4 4 !4 !4

5c 4.! 4.! !.6494*!.6494*

5s! ) 4.46!/(! )4.49469)*

5s0 04 !9.9/!)0! 04

!9.9/!)0!

e! 6.*7-4 6.*7-4 !.4*7-4)9./9)7-4

e0 !.47-4/ !.47-4/ !.6/7-4!.6/7-4


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mu!max 4.9 4.9 4.94.9

mu0max 4./ 4./ 4./4./

! 4.! 4.! 4.! 4.!

0 4. 4. 4. 4.

beta! 4.4 4.4 4.44.4

beta0 4.4 4.4 4.44.4

alpha! !.47-4) !.47-4) !.47-4)!.47-4)

alpha0 !.47-4) !.47-4) !.47-4)!.47-4)

e!max !.4*7-4) !.4*7-4) !.4*7-4) !.4*7-4)

e0max !.*67-4) !.*67-4) !.*67-4) !.*67-4)

7! 4.(66 4.(66 !.44444(04.90!(**

70 4.44/ 4.44/ 4.!!990/94.!!990/9

mu! 4./90*)! 4.*9)(/4.6(006/( 4.*9)(/

mu0 4.44*64)9 4.44*64)94.4(4!9(9 4.4(4!9(9

u! 4.99)*)) 4.6)9404.99)*)) 4.6)940

u0 4.44)// 4.44)//4.!49( 4.!49(

mumax 4./90*)! 4.*9)(/4.6(006/( 4.*9)(/

v! ! ! ! !


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8cs! 4.) 4.) 4.) 4.)

8cs0 4.) 4.) 4.) 4.)

v0 4.44)9( 4.44)9(4.!((60!/ 4.!((60!/

alphastar !.47-4/ !.47-4/ !.47-4/!.47-4/

5s!4 ) ) ) )

5s04 04 04 04 04

; 4.) 4.) 4.) 4.)

Calculated !alues of the DE1 !ariables for D@ .5


t 4 4 !4 !4

5c 4.! 4.! !.)99((!.)699(9

5s! ) 4.!0944/ )4.!*!6(99

5s0 04 !9.99(*! 04!9.99(*!

e! 6.*7-4 6.*7-4 !.4*7-4)!.40/7-4)

e0 !.47-4/ !.47-4/ ./407-4/./407-4/

mu!max 4.9 4.9 4.94.9

mu0max 4./ 4./ 4./4./

! 4.! 4.! 4.! 4.!

0 4. 4. 4. 4.


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beta! 4.4 4.4 4.44.4

beta0 4.4 4.4 4.44.4

alpha! !.47-4) !.47-4) !.47-4)!.47-4)

alpha0 !.47-4) !.47-4) !.47-4)!.47-4)

e!max !.4*7-4) !.4*7-4) !.4*7-4) !.4*7-4)

e0max !.*67-4) !.*67-4) !.*67-4) !.*67-4)

7! 4.(66 4.(66 !.444!0)(4.9()//0/

70 4.44/ 4.44/ 4.4*/!46/4.4*/!46/

mu! 4./90*)! 4.)96(6/4.6(*/!0) 4.)96(99(

mu0 4.44*64)9 4.44*64)94.40!!*/( 4.40!!*/(

u! 4.99)*)) 4.99*)(4.99)*)) 4.99*)(

u0 4.44)// 4.44)//4.4)4/0 4.4)4/0

mumax 4./90*)! 4.)96(6/4.6(*/!0) 4.)96(99(

v! ! ! ! !

8cs! 4.) 4.) 4.) 4.)

8cs0 4.) 4.) 4.) 4.)

v0 4.44)9( 4.44)9(4.4)0*(! 4.4)0*(!

alphastar !.47-4/ !.47-4/ !.47-4/!.47-4/


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5s!4 ) ) ) )

5s04 04 04 04 04

; 4. 4. 4. 4.

Calculated !alues of the DE1 !ariables for D@ .A



t 4 4 !4 !4

5c 4.! 4.! !.0(/)!*!.0(/)!*

5s! ) 4.64946 )4.64946

5s0 04 !9.999(!/ 04!9.999(!/

e! 6.*7-4 6.*7-4 !.4*7-4)!.4!7-4)

e0 !.47-4/ !.47-4/ 0.!/*7-4/0.!/*7-4/

mu!max 4.9 4.9 4.94.9

mu0max 4./ 4./ 4./4./

! 4.! 4.! 4.! 4.!

0 4. 4. 4. 4.

beta! 4.4 4.4 4.44.4

beta0 4.4 4.4 4.44.4


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alpha! !.47-4) !.47-4) !.47-4)!.47-4)

alpha0 !.47-4) !.47-4) !.47-4)!.47-4)

e!max !.4*7-4) !.4*7-4) !.4*7-4) !.4*7-4)

e0max !.*67-4) !.*67-4) !.*67-4) !.*67-4)

7! 4.(66 4.(66 !.44404)!4.996!9/!

70 4.44/ 4.44/ 4.4!)4)!(4.4!)4)!(

mu! 4./90*)! 4./90*)!4.6()(/9* 4.6446!6

mu0 4.44*64)9 4.44*64)94.4460!9 4.4460!9

u! 4.99)*)) 4.9696)4*4.99)*)) 4.9696)4*

u0 4.44)// 4.44)//4.4!4!9( 4.4!4!9(

mumax 4./90*)! 4./90*)!4.6()(/9* 4.6446!6

v! ! ! ! !

8cs! 4.) 4.) 4.) 4.)

8cs0 4.) 4.) 4.) 4.)

v0 4.44)9( 4.44)9(4.4!40/) 4.4!40/)

alphastar !.47-4/ !.47-4/ !.47-4/!.47-4/

5s!4 ) ) ) )

5s04 04 04 04 04

; 4./ 4./ 4./ 4./


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Figure E.!.". %teady state )alues from the simulation of cybernetic modele-uations. oth substrates are simultaneously utilized at low dilution rates. /t

increasing dilution rates, substrate 2 is gradually re0ected in fa)or of substrate #. /t

higher dilution rates, washout occurs with the steady state )alues same as the inletconcentrations.
http://www.engin.umich.edu/dept/che/OldFiles/OldFiles/kinetics/07chap/html/culture.polhttp://www.engin.umich.edu/dept/che/OldFiles/OldFiles/kinetics/07chap/html/culture.pol


32/70

%n mar"ed contrast to the batch culture results ofsequential utilization of the two substrates, thecontinuous culture simulations &and theexperimental data+ show the simultaneousutilization of both the substrates at low dilutionrates. 2t increasing dilution rates, the second &less

preferred or lower growth rate supporting+substrate is not consumed completely or is re$ectedin favor of the preferred &i.e. faster growth ratesupporting+ substrate. 2t much higher growthrates, the washout steady state is observed withthe two substrates and the cell mass reaching asteady state that is the same as their inletconcentration.

2.7.4.& Multiple MetabolicPathwa"s in -east0

The brewerHs or ba"erHs yeast, Saccha"omycesce"evisiae, presents an interesting example of thecybernetic ob$ective i.e. maximization of the cellgrowth rate through preferential utilization of asubstrate or in this case a metabolic pathway over

the others. The yeast cells have different pathwaysfor consuming glucose:

$1% &lucose !e"mentative #ath'ay, which may berepresented by the overall chemical equation, ifglucose consumption for cell growth is ignored:

&.!.".21'

The Monod growth parameters for this growthprocess are:


33/70

(2) Ethanol oxidative pathway, with its o)erallchemical reaction &ignoring cell growth'$

&.!.".2#'

can also occur if ethanol and oxygen are both present

in the culture medium.

The Monod growth parameters for this third growthprocess are:

and

$(%&lucose oxidative #ath'ay, which is of coursepossible only in the presence of oxygen, againignoring the glucose consumption for cell growth,the overall chemical reaction of this pathway can berepresented as:

&.!.".22'

The Monod growth parameters for this secondgrowth process are:


34/70

)y#othetical *uestion: %n a typical brewingexperiment, if glucose and oxygen are both presentin the culture medium, are both pathways usedsimultaneously or is one pathway preferentiallyutilized by the yeast, and if latter, which pathway ispreferredJ

Iumerous brewers routinely ferment glucose toethanol using yeast cells, without ta"ing any specialprecautions to eliminate oxygen from the culturemedium. These fermentations are successful &inproducing ethanol+ because cells preferentially usethe !aste"fermentative pathway and do notproducethe enzymes needed for slo'e"oxidativeconsumption of glucose even if oxygen is present inculture medium, until almost all the glucose hasbeen fermented to ethanol. 2fter all glucose isfermented, it will be necessary to stop the batchfermentation to avoid the oxidative consumption ofethanol, which will occur in a subsequent or diauxicgrowth phase if oxygen is present.

The cybernetic model equations introduced earlierpredicts the diauxic growth of yeast on glucose andethanol in aerobic cultures, with small modificationsto incorporate the specific case of ethanolgeneration from the fermentative pathway. Thefurther modified cybernetic model equations for theyeast growth metabolism &to include the dynamicsof intracellular storage carbohydrates, trehalose andglycogen, represented as 5T+ from Kones andompala &!999+ are given below for both batch &;> 4+ and continuous cultures.

&'.(.).0*+


35/70

&'.(.).0)+

&'.(.).0+

&'.(.).0/+

&'.(.).0(+

&'.(.).06+

he modified onod growth rates along the indi)idual

metabolic pathways are$

&'.(.).09+

&'.(.).*4+

&'.(.).*!+


36/70

5ybernetic variables uiand viare determined asbefore:

&'.(.).*0+

The symbols !and +represent the stoichiometricconstants for the production of ethanol,consumption of oxygen and the storagecarbohydrates respectively.

The cybernetic model for yeast metabolism predictsthe diauxic growth phases in the aerobic growth ofyeast on glucose in batch cultures &with > 4 and

high values of -a+. The model parameters werechosen to fit the experimental data from vonMeyenburg &!9/9+ and are partially listed earlierwith discussions on the three metabolic pathways..

P+,-M/) code


3ariable initial value minimal value maximal

value final value

t 4 4 04 04

5c 4.! 4.! 0//9.*(6)0//9.*(6)

5g ) -!./7@4) )-!./7@4)

5e 04 04 (94.0()*(94.0()*

5o 4. 4.4/*/)* 4.4.4/*/)*

e! 6.*7-4 6.*7-4 4.*/*!)6(4.*4/)0))

e0 !.47-4/ !.47-4/ 4.!4(949*4.!44!!9


37/70

e* !.47-4) !.47-4) 4.!0)()*)4.46!))!)

5t 4 4 4.!0!449).(697-4(

beta! 4.( 4.( 4.( 4.(

gam* 4.6 4.6 4.64.6

mu!max 4.)) 4.)) 4.))4.))

o0 4.4! 4.4! 4.4!4.4!

beta0 4.( 4.( 4.( 4.(

beta* 4.( 4.( 4.( 4.(

mu0max 4.!9 4.!9 4.!94.!9

mu*max 4.*/ 4.*/ 4.*/4.*/

alpha* 4.* 4.* 4.* 4.*

o* 0.0 0.0 0.0 0.0

alpha! 4.* 4.* 4.* 4.*

alpha0 4.* 4.* 4.* 4.*

e*max 4.06*4!69 4.06*4!694.06*4!69 4.06*4!69

e!max 4.0/*!(9 4.0/*!(94.0/*!(9 4.0/*!(9

e0max 4.**(4(6( 4.**(4(6(4.**(4(6( 4.**(4(6(

7* *.**7-4) *.**7-4)4.))4)6 4.06((9/

7! *.!)7-4) *.!)7-4)!.*64)/() !.!/))!06


38/70

70 0.9/(7-4/ 0.9/(7-4/4.*04!*46 4.*!!**

! 4.4 4.4 4.44.4

0 4.4! 4.4! 4.4!4.4!

* 4.44! 4.44! 4.44!4.44!

mu! !.*(!7-4) !.*(!7-4)!.!96466 4.!0*)*0

mu0 .0*7-4( .0*7-4(4.40/6! 4.4!!)

mu* !.!((7-4) -!.!00/6964.*4))/0* -4.44!*90

mumax !.*(!7-4) !.*(!7-4)!.!96466 4.!0*)*0

u! 4.*/(*/ 4.0/4*099.64!()( 4.9!!)(()

u0 4.440!/09 4.440!/094.*6**(9( 4.49!44)6

8! 4.!/ 4.!/ 4.!/4.!/

80 4.( 4.( 4.(4.(

8* 4./ 4./ 4./ 4./

u* 4.)/!!4!! -9.!6!0)4.)/!!4!! -4.440)600

alphastar 4.! 4.! 4.!

4.!

5g4 !4 !4 !4 !4

5s04 04 04 04 04

; 4 4 4 4


39/70

v* 4.6946*( -4.9*(49464.6946*( -4.440(0**

v! ! ! ! !

v0 4.44)4096 4.44)4096

4.!*9)*! 4.4996)*0

sigmamuv 0.*607-4) 0.*607-4)0.0!9) 4.!()))

d5c 0.*607-4 0.*607-4!*6!.06! !*6!.06!

gam! !4 !4 !4!4

gam0 !4 !4 !4

!4

d5t 6.4907-4 -4.66)9!4)4.!!)*/( -0.7-4(

phi! 4.)6 4.)6 4.)64.)6

phi0 0 0 0 0

phi* ! ! ! !

phi) 4.9 4.9 4.9 4.9

"la !444 !444 !444!444

5ostar 4.! 4.! 4.! 4.!

+DE 2eport 32K*456


d5c

;?&5g4 - 5g+ - &mu!?v!8!@mu*?v*8*+?5c - phi)?&5t?d5c @ 5c?d5t+


40/70

-;?5e @ & phi!?mu!?v!8! -mu0?v080+ ? 5c

"la ? &5ostar - 5o+ -&phi0?mu0?v080 @ phi*?mu*?v*8*+ ? 5c

alpha!?5g&!@5g+?u! -& beta!@ sigmamuv+ ? e! @ alphastar

alpha0?5e&0@5e+?u0 -& beta0@ sigmamuv+ ?e0 @ alphastar

alpha*?5g&*@5g+?u* -& beta*@ sigmamuv+ ?e* @ alphastar

d5t


.(

.6

.))

.4!

.(

.(

.!9

.*/

.*

0.0

.*

.*

alpha*& mu*max@ beta*+



41/70


e*e*max

e!e!max

e0e0max

.4

.4!

.44!

mu!max?7!?5g&!@5g+

mu0max?70?5e&0@5e+ ? 5o&o0 @5o+

mu*max?7*?5e&*@5g+ ? 5o&o* @5o+

if &mu!Emu0+ then &if &mu!Emu*+then &mu!+ else &mu*++ else &if &mu0Emu*+ then&mu0+ else&mu*++

mu!&mu!@mu0@mu*+

mu0&mu!@mu0@mu*+

.!/

.(

./4

mu*&mu!@mu0@mu*+

.!

!4

04

4

mu*mumax

mu! mumax


42/70

mu0mumax

mu!?v!@mu0?v0@mu*?v*

&sigmamuv - ;+?5c

!4

!4

gam*?mu*?v* -&gam!?mu!?v!@gam0?mu0?v0+?5t -sigmamuv?5t

.)6

0

!

.9


43/70

igure ). 5ybernetic model simulations andexperimental data from von Meyenburg &!9/9+ forcell mass, glucose, ethanol concentrations inaerobic batch culture of Saccha"omycesce"evisiae.

;uring the first growth phase, the yeast cells clearlyprefer the faster fermentative metabolism andignore or repress the oxidative metabolism. Thischoice of the fermentative pathway can beconcluded from &!+ the growth rate &the slope of asemi-long plot of cell mass+ during the first growthphase or more easily &0+ the accumulation of thefermentation product, ethanol. 2fter glucose iscompletely fermented, the presence of oxygen

enables further growth of yeast cells in a second ordiauxic growth phase using ethanol oxidativepathway.

The yeast cybernetic model equations forcontinuous or chemostat cultures can be simulatedto predict the C5rabtree effectC of preferentialutilization of glucose oxidative pathway during thelow dilution rates, followed by switch to thefermentative pathway at higher dilution rates. Theoxidative consumption of glucose, which is notutilized in the batch aerobic cultures, is thepreferred pathway for glucose consumption at thelow dilution rates, as seen from the high cell massyields in igure and absence of any ethanolproduction. 2t higher dilution rates, the utilizationof glucose fermentative pathway is seen both in lowcell mass yield in igure as well as the productionof ethanol &data not shown+.

#e can use the earlier program, and set differentvalues for ; to get various concentration plots ofthe cell and the substrates. Diven are the tables of

values for ; > .0, .*, .)

P+,-M/) /B,E0

Calculated !alues of the DE1 !ariables 3D @ .

>6


44/70


t 4 4 04 04

5c 4.! 4.49/*)/9 *(.*0!(

*(.*0!(

5g ) -004.(4*)) /.!004*4*-004.(4*))

5e 04 /.*0*040 !!4.)6(6(!!4.)6(6(

5o 4. 4.499)06* 4.4.499)06*

e! 6.*7-4 6.*7-4 4.*4(6)(

4.*490(

e0 !.47-4/ !.47-4/ 4.!466*(4.!460*9

e* !.47-4) !.47-4) 4.!40!9*4.46!!)!)

5t 4 4 4.4*/490!.*67-4/

beta! 4.( 4.( 4.( 4.(

gam* 4.6 4.6 4.64.6

mu!max 4.)) 4.)) 4.))4.))

o0 4.4! 4.4! 4.4!4.4!

beta0 4.( 4.( 4.( 4.(

beta* 4.( 4.( 4.( 4.(

mu0max 4.!9 4.!9 4.!94.!9

mu*max 4.*/ 4.*/ 4.*/4.*/


45/70

alpha* 4.* 4.* 4.* 4.*

o* 0.0 0.0 0.0 0.0

alpha! 4.* 4.* 4.* 4.*

alpha0 4.* 4.* 4.* 4.*

e*max 4.06*4!69 4.06*4!694.06*4!69 4.06*4!69

e!max 4.0/*!(9 4.0/*!(94.0/*!(9 4.0/*!(9

e0max 4.**(4(6( 4.**(4(6(4.**(4(6( 4.**(4(6(

7* *.**7-4) *.**7-4)4.*/!)9(( 4.06//996

7! *.!)7-4) *.!)7-4)!.!/96)6! !.!/000(

70 0.9/(7-4/ 0.9/(7-4/4.*006() 4.*!*9))*

! 4.4 4.4 4.44.4

0 4.4! 4.4! 4.4!4.4!

* 4.44! 4.44! 4.44!4.44!

mu! !.*(!7-4) !.*(!7-4)4.(60)/) 4.!!/09

mu0 .0*7-4( .0*7-4(4.4/6*( 4.4)!9*

mu* !.!((7-4) -4.!/!/)0/4.!!9!0 -4.4400*)0

mumax !.*(!7-4) !.*(!7-4)4.(60)/) 4.!!/09

u! 4.*/(*/ 4.*/(*/ !.0*!(*/!4.94(646


46/70

u0 4.440!/09 4.440!/094.!!060/ 4.49/!6

8! 4.!/ 4.!/ 4.!/4.!/

80 4.( 4.( 4.(4.(

8* 4./ 4./ 4./ 4./

u* 4.)/!!4!! -4.*))*!6(4.)/!!4!! -4.44*9/)*

alphastar 4.! 4.! 4.!4.!

5g4 !4 !4 !4 !4

5s04 04 04 04 04

; 4.0 4.0 4.0 4.0

v* 4.6946*( -4.0(9*9*4.6946*( -4.44)*//9

v! ! ! ! !

v0 4.44)4096 4.44)40964.!*064( 4.!490)!

sigmamuv 0.*607-4) 0.*607-4)4./060/0( 4.!(*(/

d5c -4.4!99(/0 -4.4!99(/0!!.6)4!* !!.6)4!*

gam! !4 !4 !4!4

gam0 !4 !4 !4!4

d5t 6.4907-4 -4.!9(6*//4.4!(940 -).9)/7-46

phi! 4.)6 4.)6 4.)64.)6

phi0 0 0 0 0


47/70

phi* ! ! ! !

phi) 4.9 4.9 4.94.9

"la !444 !444 !444

!444

5ostar 4.! 4.! 4.! 4.!

Calculated !alues of the DE1 !ariables 3D @ .&6


t 4 4 04 04

5c 4.! 4.49!)9!6 ).6)60)4/).6)60)4/

5g ) -!9./9*6*/ (./!/464!-!9./9*6*/

5e 04 0./*/(9)! 04!).!(/)(!

5o 4. 4.496*6/ 4.4.49990/(

e! 6.*7-4 6.*7-4 4.*4(0/*4.*4//*99

e0 !.47-4/ !.47-4/ 4.!!4*!)*4.!4*)6

e* !.47-4) !.47-4) 4.!0)64!4.464)!(

5t 4 4 4.4!9(0.94(7-4/

beta! 4.( 4.( 4.( 4.(

gam* 4.6 4.6 4.64.6


48/70

mu!max 4.)) 4.)) 4.))4.))

o0 4.4! 4.4! 4.4!4.4!

beta0 4.( 4.( 4.( 4.(

beta* 4.( 4.( 4.( 4.(

mu0max 4.!9 4.!9 4.!94.!9

mu*max 4.*/ 4.*/ 4.*/4.*/

alpha* 4.* 4.* 4.* 4.*

o* 0.0 0.0 0.0 0.0

alpha! 4.* 4.* 4.* 4.*

alpha0 4.* 4.* 4.* 4.*

e*max 4.06*4!69 4.06*4!694.06*4!69 4.06*4!69

e!max 4.0/*!(9 4.0/*!(94.0/*!(9 4.0/*!(9

e0max 4.**(4(6( 4.**(4(6(4.**(4(6( 4.**(4(6(

7* *.**7-4) *.**7-4) 4.))4)494.06)!)!(

7! *.!)7-4) *.!)7-4)!.!/((** !.!/0*!/

70 0.9/(7-4/ 0.9/(7-4/4.*0(0/4( 4.*!0**9

! 4.4 4.4 4.44.4

0 4.4! 4.4! 4.4!4.4!


49/70

* 4.44! 4.44! 4.44!4.44!

mu! !.*(!7-4) !.*(!7-4)4.!(/4/ 4.!)44/9

mu0 .0*7-4( .0*7-4(4.4/*9(9 4.4*9)!

mu* !.!((7-4) -4.4(!6064.//09(( -4.44*!99)

mumax !.*(!7-4) !.*(!7-4)4.//09(( 4.!)44/9

u! 4.*/(*/ 4.0/0/60!.4*96!00 4.9!4!!!

u0 4.440!/09 4.440!/094.!46(9)* 4.49!)!

8! 4.!/ 4.!/ 4.!/4.!/

80 4.( 4.( 4.(4.(

8* 4./ 4./ 4./ 4./

u* 4.)/!!4!! -4.!)6/4/

4./(0*!4) -4.44//!

alphastar 4.! 4.! 4.!4.!

5g4 !4 !4 !4 !4

5s04 04 04 04 04

; 4.* 4.* 4.* 4.*

v* 4.6946*( -4.!)09!/( !-4.44/00))

v! ! 4.*94(!/0 !!

v0 4.44)4096 4.44)40964.!/(09( 4.!4)9)*!


50/70

sigmamuv 0.*607-4) 0.*607-4)4./64)0( 4.!9/6(/

d5c -4.4099(/0 -4.4099(/0!.4/496* !.4/496*

gam! !4 !4 !4!4

gam0 !4 !4 !4!4

d5t 6.4907-4 -4.009**64.*(!6/6 -/.6*)7-4(

phi! 4.)6 4.)6 4.)64.)6

phi0 0 0 0 0

phi* ! ! ! !

phi) 4.9 4.9 4.94.9

"la !444 !444 !444!444

5ostar 4.! 4.! 4.! 4.!

Calculated !alues of the DE1 !ariables 3D @ .56


t 4 4 04 04

5c 4.! 4.46*9)*) 4./)0*4/*4./)0*4/*

5g ) ) 6.()9/()*/.409*09

5e 04 4.9*06(6 04!.94!9*0


51/70

5o 4. 4.49999 4.4.49999

e! 6.*7-4 6.*7-4 4.*4)(!6!4.*4)!)(

e0 !.47-4/ !.47-4/ 4.!4/(49!4.!4/))!

e* !.47-4) !.47-4) 4.46)9*4.46*!/6/

5t 4 4 /.!/!7-4.6(7-4(

beta! 4.( 4.( 4.( 4.(

gam* 4.6 4.6 4.6

4.6

mu!max 4.)) 4.)) 4.))4.))

o0 4.4! 4.4! 4.4!4.4!

beta0 4.( 4.( 4.( 4.(

beta* 4.( 4.( 4.( 4.(

mu0max 4.!9 4.!9 4.!94.!9

mu*max 4.*/ 4.*/ 4.*/4.*/

alpha* 4.* 4.* 4.* 4.*

o* 0.0 0.0 0.0 0.0

alpha! 4.* 4.* 4.* 4.*

alpha0 4.* 4.* 4.* 4.*

e*max 4.06*4!69 4.06*4!694.06*4!69 4.06*4!69

e!max 4.0/*!(9 4.0/*!(94.0/*!(9 4.0/*!(9


52/70

e0max 4.**(4(6( 4.**(4(6(4.**(4(6( 4.**(4(6(

7* *.**7-4) *.**7-4)4.*44!4*0 4.09*6/0*

7! *.!)7-4) *.!)7-4)!.!(9066 !.!(6/

70 0.9/(7-4/ 0.9/(7-4/4.*!/(4) 4.*!(6(0

! 4.4 4.4 4.44.4

0 4.4! 4.4! 4.4!4.4!

* 4.44! 4.44! 4.44!4.44!

mu! !.*(!7-4) !.*(!7-4)4.4/)946 4.4)*!*

mu0 .0*7-4( .0*7-4(4.4)!! 4.4)09*

mu* !.!((7-4) !.!((7-4)4.44666/( 4.44!)4

mumax !.*(!7-4) !.*(!7-4)4.4/)946 4.4)*!*

u! 4.*/(*/ 4.*/(*/4.94*0//* 4.94409

u0 4.440!/09 4.440!/094.49(9(0( 4.49/66!

8! 4.!/ 4.!/ 4.!/4.!/

80 4.( 4.( 4.(4.(

8* 4./ 4./ 4./ 4./

u* 4.)/!!4!! 9.4!*7-4)4.)/!!4!! 4.4409


53/70

alphastar 4.! 4.! 4.!4.!

5g4 !4 !4 !4 !4

5s04 04 04 04 04

; 4.) 4.) 4.) 4.)

v* 4.6946*( 9.9(67-4)4.6946*( 4.4406(/

v! ! ! ! !

v0 4.44)4096 4.44)40964.!!00009 4.!4(60*

sigmamuv 0.*607-4) 0.*607-4)4.!0!9(! 4.!4!906

d5c -4.4*99(/0 -4.4*99(/04.4(4((( 4.4(4(((

gam! !4 !4 !4!4

gam0 !4 !4 !4!4

d5t 6.4907-4 -.0607-4 !.!!!7-4) 0.4007-4(

phi! 4.)6 4.)6 4.)64.)6

phi0 0 0 0 0

phi* ! ! ! !

phi) 4.9 4.9 4.94.9

"la !444 !444 !444!444

5ostar 4.! 4.! 4.! 4.!


54/70

Figure . (hemostat or continuous culture steady state

data on cell mass yield on glucose

The transition from the oxidative to fermentativepathways occurs either gradually or abruptly,depending on other culture conditions, such as

oxygen supply rates, controlled mainly by impelleragitation rates, which are different in theseexperimental studies.

2.7.4.4 Metabolic +scillations :C"bernetic Model

2t the intermediate dilution rates, as the cells arechanging from oxidative to fermentativemetabolism, yeast cells can exhibit s#ontaneousmetabolic oscillations over a range of operatingconditions, such as the agitation rate or oxygenmass transfer rate. 1everal experimentalists havedocumented these oscillations in aerobic continuouscultures of yeast on glucose. The figure from Aorroet al. &!966+ shows the sustained oscillations in all


55/70

the metaboliteconcentrationsmeasured inthe continuousbioreactor. Thetop panel

shows theonlinemeasurementtrace ofdissolvedoxygenconcentrationin the culturemedium, whichis most readilyobtained. Thesubsequentpanels showoff-linemeasurements

of ethanol, glucose, intracellular storagecarbohydrates &trehalose and glycogen+, mediumpG, and cell number concentration &Lml+. Thesespontaneous oscillations change in shape, periodand amplitude as the bioreactor operatingconditions of dilution rate and agitation rate arevaried within their oscillatory ranges. 2s theseoperating conditions are varied outside theiroscillatory range, the oscillations die down to eitheroxidative or fermentative consumption of glucose.

The metabolic oscillations can also be predicted bythe yeast cybernetic model equations given above&Kones and ompala, !999+ but requires the use ofa stiff ;7 solver. The simulations were conductedon the Ner"eley Madonna software at the operatingconditions of dilution rate of 4.!/ hr-!and oxygenmass transfer rate -a&strongly affected by theagitation rate+ of *44 hr-!. The parameter valuesfor the different model constants are listed in theTable '.(.).!.

2t different operating conditions, the cyberneticmodel predicts also the experimentally observedtrends in the shape and period of oscillations as wellas the damping of the oscillations to either thefermentative or the oxidative consumption outsidethe range of oscillatory conditions.


56/70

P+,-M/) /B,E0


t 4 4 04 04

5c 4.! 4.! !64!.0)*!!64!.0)*!

5g 4.! -!.!!7@4) 4.!-!.!!7@4)

5e 4.! 4.! *06.!/*06.!/

5o !.0 4.4*/ !.04.4*/

e! 6.*7-4 6.*7-4 4.*4((!*4.*4(9*!

e0 !.47-4) !.47-4) 4.!9!)004.!4*))6

e* !.47-4) !.47-4) 4.!/!0*94.46!/*09

5t 4.! 0.4007-4( 4.!0.4007-4(

beta! 4.( 4.( 4.( 4.(

gam* 4.6 4.6 4.64.6

mu!max 4.)) 4.)) 4.))4.))

o0 4.4! 4.4! 4.4!4.4!

beta0 4.( 4.( 4.( 4.(

beta* 4.( 4.( 4.( 4.(

mu0max 4.!9 4.!9 4.!94.!9


57/70

mu*max 4.*/ 4.*/ 4.*/4.*/

alpha* 4.* 4.* 4.* 4.*

o* 0.0 0.0 0.0 0.0

alpha! 4.* 4.* 4.* 4.*

alpha0 4.* 4.* 4.* 4.*

e*max 4.06*4!69 4.06*4!694.06*4!69 4.06*4!69

e!max 4.0/*!(9 4.0/*!(94.0/*!(9 4.0/*!(9

e0max 4.**(4(6( 4.**(4(6(4.**(4(6( 4.**(4(6(

7* *.**7-4) *.**7-4)4.440** 4.066)*/*

7! *.!)7-4) *.!)7-4)!.!/9*4(( !.!/66*9

70 0.9/(7-4) 0.9/(7-4)4.)(!))() 4.*4/9!/

! 4.4 4.4 4.44.4

0 4.4! 4.4! 4.4!4.4!

* 4.44! 4.44! 4.44!4.44!

mu! 9.007-4 -4.4940/(4.*69460 4.!)096

mu0 .4607-4 .4607-44.4(996) 4.4)!*/

mu* ).))7-4 -4.49/*!(4.!4*/49 -(.69*7-4)

mumax 9.007-4 9.007-44.*69460 4.!)096


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u! 4.)90/()/ 4.!(4/)((4.9!999/ 4.9!999/

u0 4.0(4/0!( -4.64)6!)4.*(6/!(9 4.46!)!/)

8! 4.!/ 4.!/ 4.!/4.!/

80 4.( 4.( 4.(4.(

8* 4./ 4./ 4./ 4./

u* 4.0*/(4*( -4.4!9!9094.6!6/00 -4.44!)!0

alphastar 4.! 4.! 4.!

4.!

5g4 !4 !4 !4 !4

5s04 04 04 04 04

; 4 4 4 4

v* 4.)64))/) -!.)4)90) !-4.44!*)6

v! ! -!.*!(0!6 !!

v0 4.)90949 4.466)9/ !4.466)9/

sigmamuv !.)!67-4) !.)!67-4)4.)(06/ 4.!6*0(

d5c !.)!67-4 !.)!67-49**./*09) 9**./*09)

gam! !4 !4 !4!4

gam0 !4 !4 !4!4

d5t -!.!(7-4) -4.46(()!4.4/0!96 -!.6*67-4(


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phi! 4.)6 4.)6 4.)64.)6

phi0 0 0 0 0

phi* ! ! ! !

phi) 4.9 4.9 4.94.9

"la *44 *44 *44*44

5ostar 4.! 4.! 4.! 4.!

Figure !. (ybernetic model simulations of metabolic

oscillations in yeast continuous cultures. he shapeand period of the oscillations in all the metabolite

concentrations agree -ualitati)ely with the

experimental data. /s the bioreactor operatingconditions ofDand kLaare changed, the shape and

period of oscillations change as well in both

experimental data and model simulations.


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/able 2.7.4.8 ?omenclature and

Parameter alues used in ModelE9uations and Simulations

Aarameter nits ;efinition 3alue

mmax,# hr4# ax. %pecific 5rowth ate for 5lucose

Fermentation1.""

mmax,2 hr4# aximum %pecific 5rowth ate for Ethanol

6xidation

1.#7

mmax,3 hr4# aximum %pecific 5rowth ate for 5lucose

6xidation

1.38

9# g:; onod %aturation (onstant for 5lucose

Fermentation

1.1

92 g:; onod %aturation (onstant for Ethanol 6xidation 1.1#

93 g:; onod %aturation (onstant for 5lucose 6xidation 1.11#

0 gF 6xygen %aturation (onstant for Ethanol 6xidation 4.4!

* gF 6xygen %aturation (onstant for 5lucose 6xidation0.0

8! gg


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concentrations

5T gg %ntracellular 1torage 5arbohydrate &Trehalose+5onc.

variable

e%,emax,i gg %ntracellular 7nzyme 5oncentration and itsmaximum

variable

7i - 'elative 2mount of %ntracellular 7nzyme variablemi hr ! Drowth 'ate on iHth 1ubstrate or Aathway variable

; hr ! ;ilution 'ate variable

u - 5ybernetic 3ariables 5ontrolling 7nzyme1ynthesis

variable

v - 5ybernetic 3ariables 5ontrolling 7nzyme 2ctivity variable

"Fa hr ! Mass transfer coefficient for ;issolved xygen variable

2eferences0

!. ompala, ;.1., ;. 'am"rishna, I.N. Kansenand D.T. Tsao, C%nvestigation of bacterial growth onmixed substrates. 7xperimental evaluation ofcybernetic models,C Biotechnolog" andBioengineering >$:!4))-!4 &!96/+.

0. Kones, .;. and ;.1. ompala, C5yberneticmodel of the growth dynamics of Saccha"omycesce"evisiaein batch and continuous cultures,C .Biotechnolog" 780!4-!*! &!999+.

)omework Problems Based onthis Sample Section 0

!. 1how that the enzyme modification ofthe Monod growth "inetics is capable of simulatingthe presence or absence of the initial lag phase byvarying the initial level of the intracellular enzymecontent.


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0. 1how that the enzyme modification ofMonod growth "inetics does not affect thechemostat profiles of cell mass and substrate overthe range of dilution rates as well as the washoutand optimal dilution rates.

*. 7xamine whether the order ofsubstrate preference is affected by the choice ofinitial levels of the two "ey combinations &test thefour combinations: high-high, high-low, low-high,and low-low where high represents 99O of emax,iand low represents !O of emax,i+.

). Eshe"ichia coligrows on a mixture of

three sugars: glucose, xylose and lactose. TheMonod growth rate parameters for glucose are:mmax,D> !.0 hr-!, D> 4.4! gF, 85D> 4. ggP forxylose: mmax,Q> !.4 hr-!, Q> 4.4 gF, 85Q> 4.0ggP and for lactose: mmax,F> 4.6 hr-!, F> 4.! gF,85F> 4.) gg. Gow will these sugars beconsumed in a batch bioreactor with the initialsugar concentrations of gF glucose, !4 gFxylose, and 0 gF of lactose. 7xtend thecybernetic model framewor" to address threesubstrates.

. E. coliis to be cultured in achemostat on a mixture of glucose, xylose andlactose with the feed concentrations of gFglucose, !4 gF xylose, and 0 gF of lactose. TheMonod growth rate parameters are same as theones in the problem above. #hat is the optimaldilution rate that will maximize the cell massproduction rate &;?5c+J

/. ymomonas mobilishas beenengineered to ferment pentoses li"e xylose inaddition to the common hexose, glucose. TheMonod parameters for this metabolically engineeredmicroorganism during growth on glucose are mmax>4.)4 hr-!, s> 4.! gl, 8xs> 4.!! gdw g1, and 8ps> 4.)6 g ethanol g glucose. The sameparameters for growth on xylose are mmax> 4.*4hr-!, s> 4. gl, 8xs> 4.!4 gdwg1, and 8ps>


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4.) g ethanolg xylose. These cells are grown in achemostat fermentor with the feed containing 4 glglucose and 4 gl xylose. sing the Monodchemostat equations, calculate the maximumethanol production rate possible from these cellsgrowing in a chemostat. 2ssume that the

fermentative growth on xylose gets repressed orshut off completely when the glucose concentrationexceeds 4.0 gl and that both sugars are fermentedat lower dilution rates or glucose concentration R4.0 gl. Ma"e any other assumptions as needed.

(. 1olve the above problem using thecybernetic model equations, using a > 4.444! and b> 4.4 for both "ey enzymes and without using theassumptions stated in the last two sentences of

above problem.

6. 5ontinuing with the theme of aboveproblem, a second chemostat &of the same size+ isadded in series or downstream from the firstchemostat, operated at high dilution rate, to ensurethat all of xylose is consumed in continuousculture,. #hat dilution rate should be used tomaximize the ethanol production rate from themetabolically engineering ymomonas mobilisJ sethe growth parameters from the earlier problemstatement. se the cybernetic model equations orstate your additional assumptions.

0. Saccha"omyces ce"evisiaegrows in atypical diauxic growth phenomenon on a mixture ofglucose and galactose. The Monod growth rateparameters for the fermentative growth ongalactose are assumed to be mmax> 4.)4 hr-!, s>4.! gF, 8xs> 4.! g cell mass g galactose, and8ps> 4.)( g ethanol g galactose. The growth rate

parameters for oxidative growth on galactose areassumed to be mmax> 4.** hr-!, s> 4.44! gF, *> 0. gF and 8xs> 4.6 g cell mass g galactose.1imilar parameters for growth on glucose andethanol are given in Table '.(.).!. ;etermine howthe cell mass, glucose, galactose and ethanolprofiles will be in batch culture on a mixture of !4gF glucose and 04 gF galactose, if the inoculum is


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ta"en from continuous cultures on a mixture ofglucose and galactose at a dilution rate.

1. 5ontinuing with the theme of the

above problem, S. ce"evisiaeis grown in achemostat on a mixture of glucose and galactose,with the feed concentration of each being 4 gF. %tis desired to maximize the cell mass production rate&;?5c+. #hat should be the dilution rate used inthe single chemostat, assuming a -aof !444 hr-!J#atch out for the possibility of spontaneousmetabolic oscillations.

11. 5ontinuing with the theme of the

above problem further, S. ce"evisiaeis grown in achemostat on a mixture of glucose and galactose,with the feed concentration of each being 4 gF, tomaximize the ethanol production rate &;?57+. #hatshould be the dilution rate used in the singlechemostat, assuming a -aof !44 hr-!J

!0. 1imulate the metabolic oscillations ofyeast, using the cybernetic model parameters inKones and ompala &!999+ &given in the Table

'.(.).!+ to &a+ Alot how the period of oscillationschanges with dilution rate and &b+ Alot how theperiod changes with the mass transfer coefficient,-a. se a sti!!equation solver in your simulationsto obtain the oscillations.

!*. %t has been found experimentally thatthe spontaneous metabolic oscillations incontinuous cultures of yeast S. ce"evisiaecan beavoided by adding a small amount of ethanol to thefeed stream, along with glucose feed concentrationof *4 gF. %nvestigate whether the cyberneticmodel equations given in section '.(.).) canpredict the elimination of oscillations with theinclusion of ethanol in the feed stream. #hat is thesmallest ethanol concentration that will eliminatethe oscillationsJ


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!). ;etermine the parameter sensitivity ofthe cybernetic model simulations of spontaneousmetabolic oscillations in continuous cultures ofyeast for the following assumed parameter values:a, a?, b, mmax,(, *, *and 8*. irst obtain theoscillations through numerical simulations of the

cybernetic model for any combination of bioreactoroperating parameters, and -a. Iext vary each ofthe assumed parameters to determine if and howthe shape and period of oscillations change fromthe base case.

,aborator" E;ercise0 -east

*ermentation

86 =ntroduction

%n this laboratory exercise, we will study

the growth characteristics of the yeastSaccha"omyces ce"evisiaein batch cultures. 2 lab-scale & liter+ fermentor will be used to study batchgrowth "inetics of the yeast growing on glucose asthe single carbon substrate provided in thepresence of oxygen. 2 second fermentor will alsocontain glucose as the sole carbon substrate for theyeast to utilize in the absence of oxygen. 2 thirdlab-scale fermentor will be used to observe thegrowth behavior when the cells are presented witha mixture of two carbohydrates, glucose andglycerol in the presence of oxygen. 8ou will ta"esamples from the fermentors, measure the cell

mass concentration &through optical density+ anddetermine the concentrations of glucose andethanol with spectrophotometric assay "its. #iththe accumulated results from the all the studentsover eighteen hours for the three differentfermentors, you will be able to analyze the "ineticsof cell growth, and the different patterns of multiplesubstrate utilization in batch cultures.


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-east Metabolism

Saccha"omyces ce"evisiaeuses thefollowing three ma$or pathways for growth onglucose:

!+ The fermentation of glucose, which occursprimarily when the glucose concentration is high orwhen oxygen is not available. The cells attain amaximum specific growth rate of about 4.) hr-!with a low biomass yield of 4.! g dry mass pergram glucose consumed and a high respiratory

quotient &the ratio of 50production rate to the 0consumption rate+ and a low energy yield of onlyabout 0 2TA per mole of glucose metabolized. Thestoichiometry of this reaction is

5/G!0/ --------------E 050GG@ 050@ e

where e represents chemical energy utilized in thegrowth processes.

0+ The oxidation of glucose, which predominates atglucose concentrations below 4 mgl in aerobiccultures. The cells attain a maximum specificgrowth rate of only about 4.0 hr-!with a biomassyield of about 4. g dry mass per gram glucoseconsumed, a respiratory quotient of about !, and ahigh energy yield of !/-06 2TA per mole of glucose

metabolized. The stoichiometry of this reaction is:

5/G!0/@ /0 --------------E/50@ /G0 @ e


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8ou will monitor the growth characteristicsof the yeast in the batch fermentors in three ways -by measuring the concentration of cells, andpreparing cell-free samples at hourly intervals from

each bioreactor for assaying the concentrations ofglucose and ethanol. 8east cell concentration canbe determined indirectly by measuring the opticaldensity &absorbance+ of a culture sample. 8ou willta"e a sample of the culture medium from thefermentor and read its absorbance using aspectrophotometer. p to a certain cell density, theconcentration of yeast cells &gdwl+ in the sample isproportional to the absorbance reading on thespectrophotometer. The calibration curvecorrelating cell concentration with absorbancedeviates from a linear correlation at high celldensities. Necause of this, itHs a good idea to diluteany of your high ; samples &that may be on thenon-linear portion of the curve+ by a "nown dilutionfactor to confirm that the measured ; values fallon the linear portion.

The concentration of glucose, &glycerol+ andethanol in the cell-free culture samples will beanalyzed by high performance liquidchromatography or spectrophotometric assay. Toremove the cells from a * ml sample, the cell

suspension will be centrifuged and the supernatantwill be filtered through a microfiltration syringe, andassayed for glucose and ethanol by thespectrophotometric assay.

2) Experimental Procedures

6 Determining Cell Concentration

!+ eroing the spectrophotometer. 1etthe wavelength to /*4 nm. sing the "nob on theleft, set the reading to 4O transmission when thechamber is emptyP using the "nob on the right, setthe reading to !44O transmission when the


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chamber contains a test tube with about ) ml ofpure medium.

0+ ;etermining cell concentration. irst

flush out the sample tube for your groupHsfermentor by ta"ing an 6-!4 ml sample, which youwill then discard. Ta"e another 6-!4 ml samplefrom your groupHs fermentor and gently mix. Ta"eabout ) ml from your sample tube and transfer it toa glass test tube. 5lean the outside of the test tubewith ethanol, insert it into the spectrophotometer,and record the absorbance reading. %f theabsorbance reading is greater than 4.0, a typicallimit of linear correlation between the absorbanceand cell mass concentration, dilute the sample witha "nown amount of pure medium, and measure the

absorbance again to chec" if the absorbancereading is on the linear portion of the calibrationcurve. 'ecord the time you ta"e the sample alongwith the absorbance reading in the linear range aswell as the dilution factor.

B6 Spectrophotometric ssa"s forGlucose and Ethanol

To remove the cells from a * ml sample,the culture will be centrifuged and the supernatantwill be filtered through a microfiltration syringe.The lab T2 will provide more details on the assayprocedures during the experiment.

&6 2eport : Due *ebruar" 8'# 8''7

2+ ;raw a graph of:

a+ Fogarithm of cell concentrationvs. time

b+ Dlucose concentration vs. time

c+ 7thanol concentration vs. time


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for each of the three batchfermentors.

N+ ;etermine the specific growth rate andthe yield coefficient &gram dry weight of cellsproduced per gram of carbon source consumed+ foreach growth phase in the three fermentors. &Thecalibration between the absorbance reading and thedry cell mass concentration of the yeast cells will beperformed at the end of the batch cultures andprovided in the following class period+.

5+ %nterpret these two graphs in light of thebac"ground information on yeast metabolicpathways and the CcyberneticC principle that cellschoose to grow at the fastest possible rate.1pecifically, discuss why the cell mass, glucose,&glycerol+ and ethanol concentration profiles loo" asthey do for each batch fermentor.

ogler U DurmenV 0446 niversity of Michigan

microbial growth on multiple substrates

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