Microbial growth stoichiometry “fundamentals”

Marc Deront (Sirous Ebrahimi)

Microbial Growth Stoichiometry

#2_Stoichio 1

O2

Substrate

N-source

H2O H+ HCO3-

Biomass

Product

Heat

Micro-organism

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Marc Deront (Sirous Ebrahimi)#3_qSMu 2

“Black box” microbial Growth and Product kinetics

O2

Substrate

N-source

H2O H+

Biomass

Heat

qPqHeat

qS

qN

qH2O

qH+

=qX

qCO2

qO2 Product

CO2

What we already know…

Bioprocess Engineering tools from mass balances

- Rates Ri, ri, qi

- Yields Yij , Yix

Microbial consumption or production rates, 2 cases :

1. Growth without non-catabolic product

(Anabolic + Catabolic reactions)

2. Growth with non-catabolic product

(Anabolic + Catabolic + Product reactions)

IN OUT Conversions

i i iAccumulation Ri R R

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Substrate uptake for Growth

O2

Substrate

N-source

H2O H+

Biomass

Heat

qP CatabolicqHeat

qS

qN

qH2O

qH+

=qX

qCO2

qO2

Catabolic

Product

CO2

Growth without non-catabolic product (Anabolic + Catabolic reactions)

max max max

Dx Nx Ax

2 3 max max

Qx Gx

1 1 1elec donor ( )Nsource elec acceptor 1 C _molebiomass

Y Y Y

1 1(...)H O (...)HCO (...)H Heat Gibbsenergy

Y Y

Example: Aerobic growth on oxalate Growth reaction

- 5.815C2O42 (oxalate) - 0.2NH4

+ -1.8575O2 - 0.8H+

5.415H2O +1C1H1.8O0.5N0.2+10.63 HCO3-

qS Substrate uptake rate is linked to

growth rate µ by stoichiometric coefficient

of growth reaction

S growth

SX

qY

, max

1

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Substrate uptake for maintenance

Energy required for maintenance related processes …

Substrate is catabolized at mS rate in catabolic reaction

mole substrate catabolized for maintenance

mS = ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶ ̶C-mole of biomass . hr

,S main sq m

For microbial growth, substrate is

used for anabolism and maintenance

Thus:

Only growth

S growth

SX

qY

, max

1

2 3

(...)elec donor (...)elec acceptor (...)oxyd.elec donor (...)reduc.elec acceptor

(...)H O (...)HCO (...)H (...)Heat (...)Gibbsenergy

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qS and relationship, Herbert-Pirt equation

1. Substrate uptake for growth Global growth reaction

1. Substrate uptake for maintenance Catabolic reaction

Herbert-Pirt equation

YSXmax = Theoritical growth yield

YSXObs = Observed growth yield

S growth

SX

qY

, max

1

,S main sq m

1S Smax

SX

q mY

, , ,( ) ( ) ( )Obss total S S growth S mainq q q q

,

,

;1 .

; ANDHigh

Obs Obs

Obs

Obs Obs

SX SX SX

S S max SSX

S S growth max

S S growth SX SX

Y Y Yq q m

Y

m qq q Y Y

2 3

(...)elec donor (...)elec acceptor (...)oxyd.elec donor (...)reduc.elec acceptor

(...)H O (...)HCO (...)H (...)Heat (...)Gibbsenergy

max max max

Dx Nx Ax

2 3 max max

Qx Gx

1 1 1elec donor ( )Nsource elec acceptor 1C _molebiomass

Y Y Y

1 1(...)H O (...)HCO (...)H Heat Gibbsenergy

Y Y

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Only a fraction of substrate intake (– qS) [minus substrate required

for maintenance mS] is available for making new biomass with a

yield YSXmax

No growth, Only maintenance: - qS = mS (if -qS < mS then Decay!)

mS and YSXmax are model

parameters which depend on:

• Microorganism

• C-source

• e_donor

• e_acceptor

• Temperature

qS and Herbert – Pirt relationship for substrate

( )max

S s SXq m Y

(qS)

0

mS

1YSX

max

Herbert-Pirt

for substrate

1maxS SSX

q mY

or

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Substrate distribution (Maintenance)

S

S

m

q

0

1

high

most substrate

to growth

low

most substrate

to maintenance

Fraction to

maintenance

Fraction for Maintenance mS / (-qS)

S S

SSmax

SX

S S

S S

m mThus

1qm

Y

m mIfμ 0; 1 AND if μ»0; 0

-q -q

1

1

1 1

Obs

SXss

smax max

sx sx

s

Obs max

SX SX

Ymq

mY Y

mor

Y Y

YSXObs is the observed (variable) biomass growth yield

YSXMax is the theoretical biomass growth yield

maxObs

SX SXY YmaxObs

SX SXY Y


1S Smax

SX

q mY

Substrate

Growth

MaintenancemS

-qS

1max

SXY

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Substrate distribution (Growth)

Fraction for Growth


1S Smax

SX

q mY

Substrate

Growth

MaintenancemS

-qS

1max

SXY

Obs

SX

max

S SX

max

SX

m Obs

SX

max

S SX

ax

SX

Yμ 0; = 0 ;

-q Y

Yμ»0; = 1

-q Y

1μ

Y

1μ

Y

max max obs

SX SX SX

max

S SXSobs

SX

1 1

Y Y Y

1q Ym

Y

Therefore :

YSXObs = function of

and

All other yields are functions

of (by stoichiometry)

Obs

SX

max

SX

Y

Y

0

1

Fraction to

growth S

maxSX

Obs μ-qSX

max max maxmax1SX SX S SXS SXY

Y μ μ= = =

Y Y μ+m .Yμ +m Y

max

SX

S

1

Y

q

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Kinetic equation for (1)

General kinetic properties of = f (CS)

CS0

kS Half-velocity Cst

0.5max

max

= f (Cs) Monod equation:

CS

CS = large → max

µ

CS

0

0 Order

µ= a.Cs

-1 Order

CS

0

µ=µmax

1 Orderµ= b/Cs

Assumption: Under well chosen

conditions and nutrient medium, only

‘Single substrate limited growth’

= f (Cs) the well known Monod

Equationmax . S

S S

C

k C

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From Herbert-Pirt equation global growth

substrate consumption

With = f (Cs) Monod expression:(under single substrate limited growth)

Substrate rate becomes:

-qs

CS0

(max/Ysxmax)+mS

ms HOW ?


max

max- . S

S S

SX S S

Cq m

Y k C

max . S

S S

C

k C

max

1- S S

SX

q mY

With maintenance …

µ ≠ Monod equation

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qS = f (CS)

CS ≈ 0 qS ≈ 0

CS >> large qS ≈ qSmax

CS0

qS

KS Half-velocity cst

0.5qSmax

qSmax

qSmax , KS are model parameters:

qSmax = maximal substrate uptake rate

[kg S.kg X-1.hr-1]

KS = Half-velocity cst= inverse of substrate affinity

[kg S.m-3]


On the basis Michalis-Menten enzymatic rate:

qS substrate rate can be expressed using

an hyperbolic equation:

max . S

M S

CV V

K C

max SS S

S S

Cq q

K C

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Herbert-Pirt qS= f(µ) relationship AND hyperbolic qS = f(CS) kinetic:

minmax min max max; / S

S s s sx

S

S

S

CC m K Y

C

KC

max

0 then SS

S S

μ CIf m μ

K C

Monod equation

Some derived parameters, kd, max, Cs

min

kd=-mS.YSXmax CS

CSmin = mS.KS.YSX

max/µmax

max

•

max

max max

m

max

in

0; ( , )

;

0 for

S SX

S

S

S

S s

S

S

d

s

SX

C decay negative growth

C very h

m Y k

C

q m Yigh

q which occurs atm C


max

1S S

SX

q mY

max SS S

S S

Cq q

K C

max

max.S S

S SX

S S

q Cμ m Y

K C

max max max

S SX S S SX S S

max max max max

S S SX S SX S SX S max max

S Smax

S S SX

max max maxminS S SX S S SX SS max

S S

q Y C = μ +m Y . k +C

C q Y - m Y - m Y k 1μ = with q = μ +m

k +C Y

C μ -m Y k m Y kμ = with C =

k +C μ

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Marc Deront (Sirous Ebrahimi)

max

1S S

SX

q mY

#3_qSMu 13

Summary - Substrate uptake rate(No non-catabolic product)

MaintenanceGrowth

4 model parameters

YSXmax, mS , qS

max , KS

Some derived parameters, kd, max, Csmin

max max max max min max max; /d S SX S S SX s s s sxk m Y q m Y C m K Y

Herbert-Pirt formulation

Hyperbolic formulation

max SS S

S S

Cq q

K C

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From Herbert-Pirt equation, we know qS = f(µ) !

How to calculate : qNH4+, qH

+, qH2O, qHeat, qCO2, qO2…?

Example: Aerobic oxidation of glucose (no non-catabolic product) with:

1/YSXmax = 0.31 ; mS = 0.0015 [moleS.C_moleX-1.hr-1]

Substrate consumption rate (Herbert-Pirt): – qS = 0.31 µ + 0.0015

Consumed substrate is used for growth and to maintenance…

Growth reaction: – 0.31 C6H12O6 - 0.2 NH4+ – 0.82 O2 …

at µ rate + 1 C1H1.8O0.5N0.2 + 0.87 CO2 + 0.2 H+ + 1.28 H2O

Catabolic reaction (maintenance): – 1 C6H12O6 – 6 O2 + 6 CO2 + 6 H2O

at mS rate

qi specific rates calculations

max

1S S

SX

q mY

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Example: Aerobic oxidation of glucose (no non-catabolic product) with:

1/YSXmax = 0.31 ; mS = 0.0015 [mole S. C.moleX-1.hr-1]

Substrate consumption rate (Herbert-Pirt): – qS = 0.3125 µ + 0.0015

Growth reaction: – 0.31 C6H12O6 - 0.2 NH4+ – 0.82 O2 …

at µ rate + 1 C1H1.8O0.5N0.2 + 0.87 CO2 + 0.2 H+ + 1.28 H2O

Catabolic reaction (maintenance): – 1 C6H12O6 – 6 O2 + 6 CO2 + 6 H2O

at mS rate

Thus other Herbert-Pirt qi relations are in [mole Ci . C_moleX-1. hr-1]:

qX = (1) µ (-) qO2 = - 0.82 + (-6)*0.0015

(-) qS = (- 0.31) + (-1) 0.0015 (-) qNH4 = - 0.2

qCO2 = 0.87 + 6*0.0015

qH+ = 0.2

qH2O = 1.28 + 6*0.0015

qi specific rates calculations

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Using the above linear Herbert-Pirt qi relations …

qX = (1) µ (-) qO2 = - 0.82 + (-6)*0.0015

(-) qS = (- 0.31) + (-1) 0.0015 …

Yix biomass yields Yix = µ / qi Other Yields Yij = qj / qi

Thus observed yields depend on µ …!

Catabolic reaction Growth reaction Maintenance Hardly any maintenance

0

i = S, O2, CO2 ....

YSXmax= 1/0.31

YiX

YiXmax

X

i

C_molemole C

YS0

0

6

2.64

2mole O

mole Glucose

-0.82/-0.31

-6/-1

Yields calculations as function of µ

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Conclusion

There is only one free (CS, or qS)

- qS - qS

qi

CS

hyperbolic Herbert-Pirtsubstrate

CS

Yij

YiX

Herbert-Pirt for i

i = S, O2, CO2,

heat, etc.

2.64

6

i=S

j=O

Cmin

Kd ={

mS

1/YSXmax

KS

0.5qS

YSXmax

…

…

…

…

…

…

…

Catabolism

Growth

…

…

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Parameters are obtained by performing growth experiments in bioreactors:

- Batch

- Fed batch

- Chemostat-reactors

Batch (constant volume) where max, YSXmax

CS is high, thus qS≈ qSmax ≈ cst

≈ max ≈ cst

Exponential growth max, YSXmax can be estimated:

• CX = CX0.exp (max.t)

• Ln(CX/CX0 )=max.t

• YSXmax ≈ (CX-CX0) / (CS0 – CS)

BUT KS or mS are undetermined!

HOW TO get these 4 kinetic parameters ?

(qSmax or max, KS, YSX

max , mS)

max

1

1

obs

SX

S

SX

Yq ms

Y

max

SXCst Y

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Parameters are estimated by performing growth experiments in bioreactors:

- Batch

- Fed batch

- Chemostat

Chemostat under Steady State, for qSmax, YSX

max , KS, mS estimation by:

- Applying variation in flow through (load rate), D Dilution rate

- Waiting for “Steady State” dynamic equilibrium (CX and CS state

variables stabilization), where =D !

- Under only one limiting condition (Substrate CS)

- Measuring Substrate, Biomass concentration

- Applying mass-balance (for CX and CS)

Specific biomass and substrate rates , and qS are then estimated



max , mS)

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From measured and calculates, CX, CS, D, , and qS

qSmax and KS can be estimated by:

Linear fitting or Non-linear fitting



max , mS)

- qS

CS

qSmax

KS

½qSmax

Hyperbolic equation qS=f(CS)

max

S Sq = q . S

S S

C

k C

max max

Lineweaver-Burk linearisat

1

ion

1 1

S

S

S SSq C

k

q q

max max

Hanes-Woolf linearisation

1 . SS

S S S

S k

q

C

qC

q

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From measured and calculates, CX, CS, D, , and qS

YSXmax and KS can be estimated by qS = f(µ) linear fitting of

Herbert-Pirt Equation

Thus, from such chemostat experiments

qSmax or max, KS, YSX

max , mS can be estimated.



max , mS)

- qS

1/YSXmax

mS

max1 . S S

SX

mY

q

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Calculation of other rates qi in case of catabolic product only

(No non-catabolic product)

Anaerobic lactic acid (C3H6O3) fermentation from glucose (C6H12O6)

Available from chemostat studies: 1/YSXmax = 1.087 and mS = 0.025

From Herbert-Pirt specific consumption rate is: - qS = 1.087 . µ + 0.025

And according Growth reaction (at rate µ):

- 1.087 C6H12O6 – 0.2 NH4+ +1C1H1.8O0.5N0.2+ 1.8239 C3H6O3

+ 0.2 H+ +0.05 CO2 + 0.45 H2O

And Catabolic reaction (at rate mS):

- 1 C6H12O6 + 2 C3H6O3

Herbert-Pirt linear expressions of other qi [moleCi.C_moleX-1.hr-1]:

Glucose –qS = 1.087 + 0.025 H+ qH+ = 0.2

NH4+ –qN = 0.2 CO2 qC = 0.05

Lactate qP = 1.8239 + 20.025 Water qw = 0.45

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Calculation of other rates qi in case of only catabolic product

(No non-catabolic product)

Anaerobic fermentation from glucose (C6H12O6) to ethanol (C2H6O)

Available from chemostat studies: 1/YSXmax = 1.111and mS = 0.02

From Herbert-Pirt specific consumption rate is: - qS = 1.111 . µ + 0.02

And according Growth reaction (at rate µ):

- 1.111 C6H12O6 – 0.2 NH4+ + 1 C1H1.8O0.5N0.2 +1.8722 C2H6O

+ 0.2 H++1.9222 CO2+ 0.45 H2O

And Catabolic reaction (at rate mS):

– 1 C6H12O6 + 2 C2H6O + 2 CO2

Herbert-Pirt linear expressions of other qi [moleCi.C_moleX-1.hr-1]:

Glucose –qS = 1.111 + 0.02 H+ qH+ = 0.2

NH4+ –qN = 0.2 CO2 qC = 1.922 + 20.02

Ethanol qP = 1.8722 + 20.02 Water qw = 0.45

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Documents

Microbial growth stoichiometry “fundamentals”