Marc Deront (Sirous Ebrahimi)
Microbial Growth Stoichiometry
#2_Stoichio 1
O2
Substrate
N-source
H2O H+ HCO3-
Biomass
Product
Heat
Micro-organism
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
Marc Deront (Sirous Ebrahimi)#3_qSMu 3
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
Marc Deront (Sirous Ebrahimi)#3_qSMu 4
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
Marc Deront (Sirous Ebrahimi)#3_qSMu 5
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
Marc Deront (Sirous Ebrahimi)#3_qSMu 6
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
Marc Deront (Sirous Ebrahimi)#3_qSMu 7
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
Herbert-Pirt equation
1S Smax
SX
q mY
Substrate
Growth
MaintenancemS
-qS
1max
SXY
Marc Deront (Sirous Ebrahimi)#3_qSMu 8
Substrate distribution (Growth)
Fraction for Growth
Herbert-Pirt equation
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
Marc Deront (Sirous Ebrahimi)#3_qSMu 9
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
Marc Deront (Sirous Ebrahimi)#3_qSMu 10
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 ?
Kinetic equation for (2)
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
Marc Deront (Sirous Ebrahimi)#3_qSMu 11
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]
Kinetic equation for (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
Marc Deront (Sirous Ebrahimi)#3_qSMu 12
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
Kinetic equation for (4)
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
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
Marc Deront (Sirous Ebrahimi)#3_qSMu 14
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
Marc Deront (Sirous Ebrahimi)#3_qSMu 15
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
Marc Deront (Sirous Ebrahimi)#3_qSMu 16
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
Marc Deront (Sirous Ebrahimi)#3_qSMu 17
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
…
…
Marc Deront (Sirous Ebrahimi)#3_qSMu 18
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
Marc Deront (Sirous Ebrahimi)#3_qSMu 19
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
HOW TO get these 4 kinetic parameters ?
(qSmax or max, KS, YSX
max , mS)
Marc Deront (Sirous Ebrahimi)#3_qSMu 20
From measured and calculates, CX, CS, D, , and qS
qSmax and KS can be estimated by:
Linear fitting or Non-linear fitting
HOW TO get these 4 kinetic parameters ?
(qSmax or max, KS, YSX
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
Marc Deront (Sirous Ebrahimi)#3_qSMu 21
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.
HOW TO get these 4 kinetic parameters ?
(qSmax or max, KS, YSX
max , mS)
- qS
1/YSXmax
mS
max1 . S S
SX
mY
q
Marc Deront (Sirous Ebrahimi)#3_qSMu 22
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
Marc Deront (Sirous Ebrahimi)#3_qSMu 23
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