Exponential phase = log-phase
Maximum growth rates μmax
„midexponential“: bacteria often used for functional studies
Max growth rate -> smallest doubling time
Vmax S
v
K M  S
• Used when microbe population is constant = non-growing (or
short time spans)
• Derivable from first principles (enzyme-substrate binding
rates and equilibria expressions)
• Parameter determination methods used for Monod
calculations (i.e. Lineweaver Burke)
•Relates specific growth rate, , to substrate concentration
•Empirical---no theoretical basis—it just “fits”!
•Have to determine max and Ks in the lab
•Each  is determined for a different starting S
 max S

Ks  S
Michaelis Menten
 Kinetic expression derived
(theoretical)
 Constant enzyme pool


Monod
 Empirical expression
 Growth

Free enzymes
Non-growing microbes
 v vs. S where v is velocity
 Km is half saturation
constant

Enzyme concentration
increases with time
Relates microbial growth
rate constant to S
μ vs S
 Ks is half saturation
constant

 Parameters (vmax or μmax; Ks or Km) are determined by
linearization (e.g. Lineweaver Burke model) or nonlinear curve
fitting.
 Relationship between dependent variable and S determined
experimentally, in the lab
Range of S
Set conditions (T, chemistry, enzyme or microbe)
Measure the v or μ for each S
Plot v or μ vs. S; analyze data for parameter estimation
• Double reciprocal plot (Lineweaver Burke)
• Commonly used
• Caution that data spread are often insufficient
• Other linearization (Eadie Hofstee)
• Less used, better data spread
• Non-linear curve fitting
• More computationally intensive
• Progress-curve analysis (for substrate depletion)
• Less lab work (1 curve), more uncertainty
It applies where μ ǂ 0 -> exponential growth (μ = μmax ) + transition into stationary
 max S

Ks  S
KS is the half-saturation coefficient [mg/L]
Monod kinetics
-> “Substrate depletion kinetics”
Since
And
Then
And
dX
dS
 X   Y
dt
dt
 max S

Ks  S
Y Yield coefficients
Monod applies!!
dS X  max SX



dt
Y (K s  S)Y
 max
Where k =
dS kSX
 
Y
dt K s  S
k is the maximum substrate utilization rate [sec-1]
 KS is the half-saturation coefficient [mg/L]
 Substrate consumption rates have often been
described using ‘Monod kinetics’
-> Substrate controls
growth Kinetics
dS
kSX


dt K s  S
 S is the substrate concentration [mg/L]
 X is the biomass concentration [mg/ L]
 k is the maximum substrate utilization rate [sec-1]
 KS is the half-saturation coefficient [mg/L]
Stoichiometric Coefficients for Growth
Yield coefficients, Y, are defined based on the amount of consumption of
another material.

X   dX
dt 
YX / s 
 ds
s   dt 
Because ΔS changes with growth condition, YX/S is not a constant
S << KS
3
mixed order
2
S >> KS
1
max
, 1/hr
Stationary phase
μ=0
S, mg/L
Expontential
growth μ = μmax
1. Zero-order region,
S >> KS, the equation can be
approximated by μ = μmax
-> exponential growth
dS

 kX
dt
S << KS
2. Center region, Monod “mixed
order” kinetics must be used ->
transition from exponential
growth to stationary growth
caused by [S] limitation

3
2
S >> KS
1
max
dS
kSX

dt K s  S
3. First-order region,
S << KS, the equation can be
approximated as
μ = μmaxS/Ks
-> transition from exponential
growth to stationary growth 
caused by [S] limitation
Just before stationary phase
starts (stationary phase μ = 0)
mixed order
, 1/hr
dS kSX

dt
Ks
k is the maximum substrate utilization rate [sec-1]
 KS is the half-saturation coefficient [mg/L]
dS kSX


dt
Ks
S, mg/L
dS

 kX
dt
Three common assumptions
 Monod kinetics applies (mid range concentrations)
-> “Substrate depletion kinetics”
 First-order decay (low concentration of S,
applicable to many natural systems)
 Zero-order decay (substrate saturated) μ =μmax
-> exponential growth
Cellular growth rate
Monod approximation
dX
 μ X
dt
dX
s X
 μmax 
dt
Ks  s
s
μ  μmax 
Κs  s
Yield factor

X   dX
dt 
YX / s 
 ds
s   dt 
Substrate Utilization
YP / X

P   dP
dt 

 dX
X   dt 
ds μmax s  X
 

dt YX / s K s  s
Product Formation

(Beginning of Stationary Phase)
dP
s X
 YP / X   max 
dt
Ks  s
Rate per microbe, which depends on
Species
Substrates
Environmental factors
Total numbers of microbes
• Culture-based
(limited: 2000 species vs. 13,000 species of bacteria
in soil by DNA-based methods
•Counting
colony forming units (CFUs)
•Activity assays: need cell or biomass count to normalize
• Culture-independent
•Direct Counts
•General
fluorescent stain, like acridine orange or SYBR gold
•Counting cells in FISH assay
•Biomass
assays
•Quantification
of an element like C or N
•Chloroform fumigation / incubation or direct extraction
•Total protein or DNA
-> Why is it important to know the kinetics of the reaction in the
fermenter?
-> What is going on in a fermenter?
-> How to control the process in a fermenter?
Stochiometric Coefficients
-> Too complex !!!!
-> Blackbox effect
substrates + cells → extracellular products + more cells
( ∑S + X → ∑P + nX)
Monod’s model -> S depletion
1. Mass balance : depentend on reactor type -> S, P, X
2. Growth Kinetics: -> Monod model (substrate depleting model)
-> Describes what happens in the reactor in steady state
(constant conditions)
Primary metabolic products
Secondary metabolic products
Microbial Products
1. Growth associated products
: products appear simultaneoulsy with cells in culture
1 dP
qp 
 YP / X 
X dt
qp is the specific rate of product formation (mg product per g biomas per hours
2. Non-growth associated products
: products appear during stationary phase of batch growth
q p    cons tan t
3. Mixed-growth associated products
: products appear during slow growth and stationary phase
q p    
Biotechnological processes of growing
microorganisms in a bioreactor
Mass Balance:
Fin = Fout = 0
V= const.
Fin ≠ 0; Fout = 0
V increases
Fin = Fout ≠ 0
V = const.
1. Mass balance : depentend on reactor type -> S, P, X
2. Growth Kinetics: -> Monod model (substrate depleting model)
-> Describes what happens in the reactor in steady state
(constant conditions)
1. Mass Ballance:
Biomass:
In – Out + Reaction = Accumulation
FX0 - FX + ∫r dV = dn/dt
r = dX/dt = µ X
2. Monod Kinetics:
 max S

Ks  S
dn/dt = d(XV)/dt
dn/dt=V (dX/dt) + X (dV/dt)
3. Steady state: dX/dt = 0 (NOT for Batch reactor!!!)
Closed
Well-mixed
Constant volume
-> substrate growth limiting factor
V= const.
Mass Balance - Biomass:
Acc = dn/dt
Verbal:
n = mole
In – Out + Reaction = Accumulation
dn/dt = d(XV)/dt = (dX/dt) V
Math: 0
0
rV
dX/dt V
Rearrange:
r V
= dX/dt V
Growth
-> Substrate concentration controls growth rate
dX
 μ X  r
dt
Growth
Cellular growth rate
Monod approximation
dX
 μ X
dt
dX
s X
 μmax 
dt
Ks  s
s
μ  μmax 
Κs  s
Yield factor

X   dX
dt 
YX / s 
 ds
s   dt 
Substrate Utilization
YP / X

P   dP
dt 

 dX
X   dt 
ds μmax s  X
 

dt YX / s K s  s
Product Formation

(Beginning of Stationary Phase)
dP
s X
 YP / X   max 
dt
Ks  s
Biotechnological processes of growing
microorganisms in a bioreactor
Mass Balance:
Fin = Fout = 0
V= const.
Fin ≠ 0; Fout = 0
V increases
Fin = Fout ≠ 0
V = const.