CHAPTER 3:

KINETICS OF GROWTH

IN BATCH AND

CONTINOUS CULTURE

ERT 317 BIOCHEMICAL ENGINEERING SEM 1 2012/13

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

Microbial growth is an autocatalytic reaction

The rate of growth is directly related to cell concentration

Characterized by the net specific growth rate:

nXPXS

Cells MoreProductslar ExtracelluCellsSubstrate

dt

dX

Xnet

1

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):

Microbial growth can also be described in terms of

cell number, N:

where μR is the net specific replication rate (h-1)

dgnet k

dt

dN

NR

1

Batch Growth –Determining Cell

Number Density

Hemocytometer

Direct microscopic count

Counts all cells present (viable and non-viable)

Immediate result

Agar plates

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:

t

net

net

neteXXtX

X

tXXXdt

dX

0

0

0

or ln

0at where

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:

The doubling time based on cell number is expressed

as:

maxmax

693.02ln

d

R

d

2ln'

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 energy-producing 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:

tk

SOddeXXtk

dt

dX or

Death Phase

The death or decline phase is characterized by the expression:

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’

tk

SddeNNtk

dt

dN '

or '

Death Phase

1. Cell lysis (spillage) may occur

2. Rate of cell decline is first-order

where: –kd = 1st order death rate constant,

Xs = conc. of cell at end of stationary phase

3. 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:

For 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

S

XY SX

/

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:

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

energymaintence

energygrowth

productlarextracellu

an intoonassimilati

biomass intoonassimilati SSSSS

Yield Coefficients

Yield coefficients can also be defined for other substrates or for product formation:

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)

S

PY

O

XY

SP

OX

/

2

/ 2

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:

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

X

dtdSm m/

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:

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

gXPp Ydt

dP

Xq /

1

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:

Many secondary metabolites, such as most

antibiotics (e.g. penicillin), are non-growth-

associated products

constant pq

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:

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

gpq

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 expressed by:

Both and vary with temperature according to the Arrhenius equation:

Where:

Ea =activation energy for growth ≈ 10-20 kcal/mol

Ed =activation energy for death ≈ 60-80 kcal/mol

Nkdt

dNdR

''

RTE

d

RTE

Raa AekAe

/'/'

'

R'

dk

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+, NO3-

consumes 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…)

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

For substrates:

The oxidation state of S has a large effect on 1/ YH

Rate of Heat Generation by Growth,

QGr

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