SEM 1 2012/13
CHAPTER 3:
KINETICS OF GROWTH
IN BATCH AND
CONTINOUS CULTURE
ERT 317 BIOCHEMICAL ENGINEERING
Lecture Outline


Introduction
Batch Growth Characteristics
 Growth
Stages, Effects of Environmental Conditions,
Product Formation, Mathematical Models

Continuous Growth Characteristics
 Dilution
Rate, Optimum Operation
Cell growth
Substrate  Cells  Extracellu lar Products  More Cells
 S  X   P  nX

Microbial growth is an autocatalytic reaction
 The
rate of growth is directly related to cell concentration
 Characterized by the net specific growth rate:
 net
1 dX

X dt
Cell growth

The net specific growth rate is the difference between
a gross specific growth rate (μg, h-1) and the rate of
loss of cell mass due to cell death (kd, h-1):
 net   g  kd

Microbial growth can also be described in terms of
cell number, N:
1 dN
R 
N dt

where μR is the net specific replication rate (h-1)
Batch Growth –Determining Cell
Number Density

Hemocytometer




Agar plates





Direct microscopic count
Counts all cells present (viable and non-viable)
Immediate result
Counts only living cells
Delayed result
Assumption: each viable cell will yield 1 colony
Results expressed in CFUs(colony-forming units)
Particle counters



Counts all cells present (viable and non-viable)
Suitable for discrete cells in a particulate-free medium
Can distinguish between cells of different sizes
Hemocytometer
Viable Cell Count
Coulter Particle Counter
Determining Cell Mass Concentration – Direct
Methods

Dry cell weight (DCW)
A
sample of fermentation broth is centrifuged, washed,
and dried at 80°C for 24hrs
 Off-line measurement; wet cell weights (WCW) can
performed in-process

Packed cell volume
 Like

wet cell weight, but measures cell pellet volume
Optical density (OD)
 Turbidity
–based on the absorption of light by suspended
cells in culture media
Determining Cell Mass Concentration – Indirect
Methods


In many fermentation processes, particularly with moulds,
direct methods cannot be used
Indirect methods are therefore employed, based on the
measurement of substrate consumption and/or product
formation
Intracellular components of cells such as RNA, DNA and protein
can be measured as indirect indicators of cell growth
 Concentration of RNA/cell weight varies significantly during a
batch growth cycle, while DNA and protein concentrations per
cell weight remain fairly constant, and can therefore be used
as reasonable measures of cell growth

Time-Dependent Changes in Cell
Composition and Cell Size
Azotobacter vinelandii Growth in Batch Culture
Batch Growth
Batch Growth Curve
Growth Phases
1. Lag
2. Exponential
3. Deceleration
4. Stationary
5. Death/Decline
Lag Phase

Occurs immediately after inoculation and is a
period of adaptation for the cells to their new
environment
 New
enzymes are synthesized, synthesis of other
enzymes is repressed
 Intracellular machinery adapts to the new conditions
 May be a slight increase in cell mass and volume, but
no increase in cell number
 The lag phase can be shortened by high inoculum
volume, good inoculum condition (high % of living cells),
age of inoculum, nutrient-rich medium
Influence of [Mg2+] on Lag Phase Duration in E.
aerogenes Culture
 E. aerogenes requires Mg2+ to
activate the enzyme
phosphatase, which is
required for energy
generation by the organism
 The concentration of Mg2+ in
the medium is indirectly
proportional to the duration
of the lag phase
Exponential Growth Phase

In this phase, the cells have adjusted to their new
environment
At this point the cells multiply rapidly (exponentially)
 Balanced growth –all components of a cell grow at the same
rate
 Growth rate is independent of nutrient concentration, as
nutrients are in excess
 The first order exponential growth rate expression is:

dX
  net X where X  X 0 at t  0
dt
X
ln
  nett or X  X 0 e  nett
X0
Exponential Growth Phase (cont’d)


An important parameter in the exponential phase is
the doubling time (time required to double the
microbial mass)
A graph of ln X versus t produces a straight line on a
semi-logarithmic plot:
ln 2 0.693
d 

 max

 max
The doubling time based on cell number is expressed
as:
ln 2
'
d 
R
Exponential Growth Phase (cont…)
t
Deceleration Phase

Very short phase, during which growth decelerates
due to either:
 Depletion
of one or more essential nutrients, or,
 The accumulation of toxic by-products of growth (e.g.
Ethanol in yeast fermentations)



Period of unbalanced growth: td=td’
Cells undergo internal restructuring to increase their
chances of survival
Followed quickly by the Stationary Phase
Stationary Phase


Starts at the end of the Deceleration Phase, when the
net growth rate is zero (no cell division, or growth rate
is equal to death rate)
Cells are still metabolically active, and can produce
secondary metabolites
Primary metabolites are growth-related products, while
secondary metabolites are non-growth-related
 Many antibiotics and some hormones are produced as
secondary metabolites
 Secondary metabolites are produced as a result of
metabolite deregulation

Stationary Phase (cont’d)

During this phase, one or more of the following
phenomena may occur:
 Total
cell mass concentration may stay constant, but the
number of viable cells may decrease
 Cell lysis may occur, and viable cell mass may drop. A
second growth phase may occur as cells grow on lysis
products from the dead cells (cryptic growth)
 Cells may not be growing, but may have active
metabolism to produce secondary metabolites
Stationary Phase (cont’d)

During the stationary phase, the cell catabolizes cellular
reserves for new building blocks and for energyproducing monomers


This is called endogenous metabolism
The cell must expend maintenance energy in order to stay
alive

The equation that describes the conversion of cellular mass into
energy, or the loss of cell mass due to lysis during the
stationary phase is:
dX
kd t
 kd t or X  X SOe
dt
Death Phase



The death or decline phase is characterized by the
expression:
dN
 k d' t
'
 kd t or N  N S e
dt
Where Ns is the concentration of cells at the end of
the stationary phase, and is the first-order death-rate
constant
A plot of ln N versus t yields a line of slope –kd’
Death Phase
1.
2.
Cell lysis (spillage) may occur
Rate of cell decline is first-order
where:
3.
–kd = 1st order death rate constant,
Xs = conc. of cell at end of stationary phase
Growth can be re-established by transferring to fresh media
Yield Coefficients


Growth kinetics are generally further described by
defining stoichiometrically related parameters
Yield coefficients are defined based on the amount of
consumption of a given material
 For
example, the growth yield coefficient is:
YX / S
 For
X

S
organisms growing aerobically on glucose, Yx/s is
typically 0.4 to 0.6 g/g, for most yeast and bacteria;
anaerobic growth is much less efficient
Aerobic and Anerobic Growth Yields of
S. faecalis on Glucose
Yield Coefficients

At the end of a batch growth period, there is an
apparent or observed growth yield:
S  Sassimilation  Sassimilation  Sgrowth  S maintence
into biomass

into an
extracellular
product
energy
energy
The apparent yield is not a true constant for
compounds that can be used as both a carbon and
energy source, but the true growth yield (YX/S) is
constant ΔS
Yield Coefficients

Yield coefficients can also be defined for other
substrates or for product formation:
YX / O2
YP / S

X

O2
P

S
YX/O2 is typically 0.9 to 1.4 g/g for most yeast and
bacteria, but is much lower for highly reduced
substrates (e.g. methane, CH4)
Summary of Yield Factors for Aerobic
Growth
The Maintenance Coefficient

The maintenance coefficient is used to describe the specific
rate of substrate uptake for cellular maintenance:

dS / dt m
m
X


However, during the Stationary Phase, where little external
substrate is available, endogenous metabolism of biomass
components is used for maintenance energy
Maintenance energy is the energy required to repair
damaged cellular components, to transfer nutrients and
products in and out of cells, for motility, and to adjust the
osmolarity of the cells’ interior volume
Microbial Products

Microbial products can be classified into three major
categories
Growth-associated products
 Non-growth-associated products
 Mixed-growth-associated products


Growth-associated products
These products are produced simultaneously with microbial
growth
 Specific rate of product formation is proportional to the
specific growth rate, μg
 Note that μg is not equal to μnet, the net specific growth rate,
when endogenous metabolism is occurring

Growth-Associated Products

The rate expression for product formation in
growth-associated production is:
1 dP
qp 
 YP / X  g
X dt


Where qp is the rate of product formation (h-1)
The production of a constitutive (continuously
produced, as opposed to inducible) enzyme is an
example of a growth-associated product
Non-Growth-Associated Products



Non-growth-associated product formation takes
place during the Stationary Phase, when the growth
rate is zero
Specific rate of product formation is constant:
q p    constant
Many secondary metabolites, such as most
antibiotics (e.g. penicillin), are non-growthassociated products
Mixed-Growth-Associated Products


Mixed-growth-associated product formation takes place during
the Deceleration (slow growth) and Stationary Phases
The specific rate of product formation is given by the
Luedeking-Piret equation:
q p  g  


If α= 0, the product is completely non-growth associated; If β=
0, the product is completely growth-associated
Examples: lactic acid fermentation, production of xanthan gum,
some secondary metabolites
Product Yield Coefficients (cont…)
a) Growth-associated product formation
b) Non-growth-associated product formation
c)
Mixed-growth-associated product formation
Environmental Factors


Patterns of microbial growth and product formation
are influenced by environmental factors such as
temperature, pH and dissolved oxygen concentration
(D.O.)
Microorganisms can be classified by their optimum
growth temperatures, Topt
Psychrophiles: (Topt< 20°C)
 Mesophiles: (20°C < Topt< 50°C)
 Thermophiles: (Topt> 50o°C)


As the temperature increases towards Topt, the growth
rate doubles every ~10°C
Optimum Growth Temperature
Optimum Growth Temperature
Effect of Temperature on Cell Growth

Above Topt the growth rate decreases and thermal death
may occur





The net specific replication rate for temperatures above Topt is
dN
expressed by:
'
'
dt


  R  kd N
Both  and k vary with temperature according to
the Arrhenius equation:
'
R
'
d
  Ae
'
R
 Ea / RT
k   Ae
'
d
 Ea / RT
Where:
Ea =activation energy for growth ≈ 10-20 kcal/mol
Ed =activation energy for death ≈ 60-80 kcal/mol
Arrhenius Plot of Growth Rate of E. Coli
Legend:
(●) Growth on
rich, complex
medium
(○) Growth on
glucose-mineral
salts medium
Effect of pH on Cell Growth
pH affects the activity of enzymes, and therefore
the microbial growth rate
 Acceptable pH’s for growth are typically within 1 or
2 pH units of the optimum pH
 pH range varies by organism:

 bacteria
(most) pH = 3 to 8
 yeast pH = 3 to 6
 plants pH = 5 to 6
 animals pH = 6.5 to 7.5
Effect of pH on Cell Growth



The optimal pH for growth may be different from
the optimal pH for product formation (e.g. Pichia
pastoris)
Microorganism have the ability to control pH inside
the cell, but this requires maintenance energy
pH can change due to:
 Utilization
of substrates; NH4+ releases H+, NO3consumes H+
 Production of organic acids, amino acids, CO2, bases
Effect of pH on Cell Growth (cont…)
Effect of Dissolved O2 on Cell Growth

At high cell concentrations, the rate of oxygen
consumption may exceed the rate of O2 supply



When oxygen is the rate-limiting factor, specific growth rate
varies with [DO] according to saturation (Michaelis-Menten)
kinetics
Below a critical concentration, growth approaches a
first-order rate dependence on DO (oxygen is a limiting
substrate)
Above a critical concentration, the growth rate becomes
independent of DO (oxygen is non-limiting))
Effect of Dissolved O2 on Cell Growth (cont…)
Obligate aerobic cells
Saturation kinetics
Facultative aerobic cells
Saturation kinetics
Effect of Dissolved O2 on Cell Growth

The saturated DO concentration for water at 25°C
and 1 atm is ~7 ppm
 The
presence of dissolved salts and organics can alter
the saturation value
 Increasing temperatures decrease the saturation value

The critical oxygen concentration is about 5%-10%
of the saturated DO concentration for bacteria and
yeast, and about 10%-50% of [DO]sat for moulds,
since they grow as large spheres in suspended
culture (diffusion issues)
Other Effects on Cell Growth

Dissolved CO2 can have a profound effect on the
performance of microorganisms



Very high DCO2 concentrations can be toxic to some cells
On the other hand, cells require a certain minimum DCO2 level
for proper metabolic function
Ionic strength (I); too high dissolved salts is inhibitory to
membrane function (membrane transport of nutrients, osmotic
pressure):
where :
Ci = molar concentration of ion i
Zi = ion charge
Other Effects on Cell Growth

The redox potential is an important parameter that affects the rate and
extent of many oxidative-reductive reactions


In fermentation media, the redox potential is a complex
function of DO, pH, and other ion concentrations, such as
reducing and oxidizing agents
Substrate concentrations significantly above stoichiometric requirements
are inhibitory to cellular functions
Inhibitory levels of substrates vary depending on cell type and
substrate
 Typical maximum non-inhibitory concentrations of some
nutrients are –glucose, 100 g/l; ethanol, 50 g/l for yeast, much
less for other organisms; ammonium, 5 g/l; phosphate, 10 g/l;
nitrate, 5 g/l

Heat Generation by Growth

About 40% to 50% of the energy stored in a carbon and energy
source is converted to biological energy (ATP) during aerobic
metabolism, and the rest of the energy is released as heat



For actively growing cells, the maintenance requirement is low, and heat
evolution is directly related to growth
The heat of combustion of the substrate is equal to the sum of the metabolic
heat and the heat of combustion of the cellular material:
Where ΔHS is the heat of combustion of the substrate (kJ/g substrate), ΔHC
is the heat of combustion of cells, and 1/YH is the metabolic heat evolved
per gram of cell mass produced (kJ/g cells)
Energy Balance on Microbial Utilization
of Substrate
Heat Generation by Growth


The above equation in heat generation can be
rearranged to become:
ΔHS and ΔHC can be determined from the combustion of
substrate and cells
Typical ΔHC values for bacterial cells are 20-25 kJ/g cells
 Typical values of YH are: glucose, 0.4 g/kcal; malate, 0.3
g/kcal; acetate, 0.21 g/kcal; ethanol, 0.18 g/kcal; methanol,
0.12 g/kcal; and methane, 0.061 g/kcal
 Clearly, the degree of oxidation of the substrate has a strong
effect on the amount of heat released

Heat Generation by Growth (cont…)
For substrates:
Substrate, S
∆Hs (kJ/g S)
YH (g dcw/kJ)
Glucose
15.64
0.072
Methanol
22.68
0.029
Ethanol
29.67
0.043
n-Decane
47.64
0.038
Methane
55.51
0.015
The oxidation state of S has a large effect on 1/ YH
Rate of Heat Generation by Growth,
Q
Gr

The total rate of heat evolution in a batch
fermentation is:
where: VL = liquid volume

In aerobic fermentations, the rate of metabolic heat
evolution can roughly be correlated to the rate of
oxygen uptake:
 where:
QGR is in kcal/h, and QO2 is in mM of O2/h
Heat Generation by Microbial Growth



Metabolic heat released during a fermentation can
be removed by circulating cooling water through a
cooling coil within the fermenter, or a cooling jacket
surrounding the fermenter
Temperature control is a critical limitation on reactor
design
The ability to estimate heat removal is essential to
proper reactor design
Cooling coils and Water Jacketed Fermenter