Cells Growth in Continuous Culture
Continuous culture: fresh nutrient medium is
continually supplied to a well-stirred culture and
products and cells are simultaneously withdrawn.
At steady state, concentrations of cells, products
and substrates are constant.
In batch culture: the culture environment changes
continually.
growth, product formation and substrate
utilization terminate after a certain time interval.
A continuous-culture laboratory setup
medium
Products,
cells
Ideal Chemostat
Same as perfectly mixed continuous-flow, stirred-tank
reactor (CFSTR).
- Control elements: pH, dissolved oxygen, temperature
- Fresh sterile medium is fed to the completely mixed
and aerated (if required) reactor.
- Suspension is removed at the same rate.
- Liquid volume in the reactor is kept constant.
Cell Growth in Ideal Chemostat
A material balance on the cell and substrate concentration
around the chemostat yields,
FX 0  FX  VR  g X  VR kd X  VR
1
FS0  FS  VR  g X
M
Y
dX
dt
 VR q p X
1
Yp / s
 VR
dS
dt
X /S
F is the volumetric flowrate of nutrient solution (l/h);
VR is the culture volume (l) (constant); X is the cell concentration (g/l);
P is the extracellular product (g/l);
µg and kd are growth rate and endogenous rate constant, respectively (h-1).
qp is the specific rate of extracellular product formation (g P/g cells-h)
Yp/s is the product yield coefficient (g P/g S).
Subscript 0 denotes the parameters at the feed medium.
Cell Growth in Ideal Chemostat
At steady state, X0=0, kd ≈ 0, qp=0 , Monod equation applied,
0
0
0
dX
FX 0  FX  VR  g X  VR kd X  VR
dt
0
1
FS0  FS  VR  g X
dS
 VR q p X
 VR
Yp / s
dt
M
Y
X /S
g 
m S
KS  S
1
0
Cell Growth in Ideal Chemostat
At steady state, X0=0, kd ≈ 0, qp=0 , Monod equation applied,
g  D 
m S
S
KS  S
M
K D
S
 m  D Assign. Q 6.13, qo
2
M
Ks D
X Y
( S0  S )  Y
( S0 
)
m  D
X /S
X /S
Cell Productivity: DX
Dopt
d ( DX )
 0  Dopt
dD
Ks
  m (1 
)
K s  S0
X opt  Y X / S ( S 0  K s ( K s  S 0 )  K s )
M
D=F/VR, is dilution rate (1/time), the reciprocal of residence time.
Cell Growth in Ideal Chemostat
Washed out: If D is set at a value greater than µm (D > µm),
the culture cannot reproduce quickly enough to maintain itself.
4
0.3
DX
µm = 0.2 hr-1
3.5
0.25
0.2
2.5
X
2
S
0.15
1.5
0.1
1
0.05
0.5
0
0
0.05
0.1
0.15
D (1/hr)
0.2
0
0.25
DX (g/L-hr)
S, X (g/L)
3
Determination of Monod Parameters
In Chemostat: µg=D, varying D obtains D~S
g  D 
m S
KS  S
S
K D
S
m  D
m 1
1
1


( Lineweaver - Burk)
S
KS D KS
Chemostat technique: reliable, constant environment,
operation may be difficult.
Determination of Monod Parameters
Batch: X, S, t → lnX ~ t , get µm (slope) from data
in exponential phase.
X
ln
 μnet t  μm t
X0
m S
dX
 g 
,
Xdt
KS  S
1
g
S
g
Kd  0
KS 1
1


( Lineweaver - Burk)
m S
m

KS
m

S
m
(Hanes - Woolf)
Cell Growth in Ideal Chemostat
At steady state, X0=0, kd > 0, qp=0 , Monod equation applied,
0
0
dX
FX 0  FX  VR  g X  VR kd X  VR
dt
0
1
FS0  FS  VR  g X
dS
 VR q p X
 VR
Yp / s
dt
M
Y
X /S
g 
m S
KS  S
1
0
Cells Growth in Continuous Culture
Ideal Chemostat
Endogenous metabolism (X0=0, kd > 0, qp=0) is considered
Equation summary
 net   g  k d 
m S
 kd
Ks  S
Ks ( D  k d )
 g  D  kd ; S 
m  D  kd
X  Y M (S0  S ) D
X /S
D  kd
Productivities: DX
1
AP
Y
1

M
Y
X /S
X /S
ms

D
kd
ms 
M
Y
X /S
Cells Growth in Continuous Culture
Ideal Chemostat
Endogenous metabolism (X0=0, kd > 0, qp=0) is considered
1
AP
Y
1

M
Y
X /S
ms

D
X /S
Y M and ms can be obtained from chemostat experiemnt s
X /S
X
AP
AP
by plotting1/Y
(Y

) against1/D.
X /S X /S S  S
0
kd
Then k d can be obtained from
ms 
M
Y
X /S
Cells Growth in Continuous Culture
Ideal Chemostat
Endogenous metabolism (X0=0, kd > 0, qp=0) is considered.
Ks and  m can be determined .
Ks( D  k d )
S
m  D  kd
m
1
1


S Ks( D  k d ) Ks
1
1
Plot
versus
S
(D  kd )
m
1
slope 
, y intercept  
Ks
Ks
Cell Growth in Ideal Chemostat
At steady state, X0=0, kd >0, qp>0 , Monod equation applied,
0
dX0
FX 0  FX  VR  g X  VR kd X  VR
dt
1
FS0  FS  VR  g X
M
Y
g 
m S
KS  S
0
dS
 VR q p X
 VR
Yp / s
dt
1
X /S
The mass balance on product formation :
dP
dP
V R q p X  FP  V R
 q p X  DP 
dt
dt
at steady state, DP  q p X
Cells Growth in Continuous Culture
Ideal Chemostat
Endogenous metabolism (kd > 0, qp > 0) is considered
Equation summary:
 g  D  kd
Ks ( D  kd )
S
m  D  kd
X  Y M ( S0  S )
X /S
D
YXM/ S
D kd  q p
Yp / s
DP  q p X
Productivities: DP, DX
Use graphical method to determine the optimum DP, DX.
Summary of Growth Kinetics
- Autocatalytic reaction: The rate of growth is directly related
to cell concentration
Net specific growth rate (1/time):
1 dX
net 
X dt
net   g  kd
- Cell concentration determination
- Growth patterns and kinetics in batch culture
- lag phase
X
- logrithmic or exponential growth phase: ln X  μnet t ,  d
0
- deceleration phase
- stationary phase: endogenous metabolism dX  kd X
dt
- death phase
Summary of Growth Kinetics
- Effect of factors:
- Dissolved oxygen:
oxygen consumption rate :
g X
YX / O
2
oxygen transfer rate : k L a(C * C )
- Temperature, pH, ionic strength, substrate
concentration.
- Heat evolution:
H s
1
 H c 
YX /S
YH
Summary of Growth Kinetics
- Monod equation:
g 
m S
KS  S
-Cell growth in continuous culture:
qp=0, kd ≈0, X0=0, Monod equation is applied:
g  D 
X Y
M
X /S
Productivity: DX
m S
S
KS  S
(S0  S )
X Y
X /S
Dopt   m (1 
X opt  Y
M
M
X /S
K D
S
m  D
( S0  S )  Y
M
X /S
( S0 
Ks
)
K s  S0
(S0  K s ( K s  S0 )  K s )
Ks D
)
m  D
Summary of Growth Kinetics
-Cell growth in continuous culture:
qp=0, kd >0, X0=0, Monod equation is applied:
m S
 net   g  k d 
 kd
Ks  S
Ks ( D  k d )
 g  D  kd ; S 
m  D  kd
X  Y M (S0  S ) D
X /S
D  kd
Productivities: DP, DX
1
AP
Y
1

M
Y
X /S
X /S
ms

D
Summary of Growth Kinetics
-Cell growth in continuous culture:
qp>0, kd >0, X0=0, Monod equation is applied:
DP  q p X
X  Y M ( S0  S )
X /S
D
YXM/ S
D kd  q p
Yp / s
 g  D  kd
Ks ( D  kd )
S
m  D  kd
Productivities: DP, DX